on local galois representations attached to automorphic forms - IWR

ON LOCAL GALOIS REPRESENTATIONS
ATTACHED TO AUTOMORPHIC FORMS
A Thesis
Submitted to the
Tata Institute of Fundamental Research, Mumbai
for the degree of Doctor of Philosophy
in Mathematics
by
VG NARASIMHA KUMAR CH
School of Mathematics
Tata Institute of Fundamental Research
Mumbai
November, 2010

DECLARATION
This thesis is a presentation of my original research work. Wherever contributions
of others are involved, every effort is made to indicate this clearly, with due reference
to the literature, and acknowledgement of collaborative research and discussions.
The work was done under the guidance of Professor Eknath Ghate, at the Tata
Institute of Fundamental Research, Mumbai.
[VG Narasimha Kumar CH]
In my capacity as supervisor of the candidate’s thesis, I certify that the above
statements are true to the best of my knowledge.
[Professor Eknath Ghate]
Date:

Contents
1 Synopsis 11
1.1 (p, p)-Galois representations attached to automorphic forms on GL n . . . 11
1.2 2-adic Hida theory and applications . . . . . . . . . . . . . . . . . . . . . 14
2 Introduction 17
3 Modular forms and Hecke algebras 25
3.1 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Hecke operators for Γ 1 (N) . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Oldforms and Newforms . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.3 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Galois Representations and Automorphic Forms on GL n 31
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 p-adic Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 Fontaine’s rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.2 Newton and Hodge numbers . . . . . . . . . . . . . . . . . . . . . 34
4.2.3 Potentially semistable representations . . . . . . . . . . . . . . . . 35
4.2.4 Weil-Deligne representations . . . . . . . . . . . . . . . . . . . . . 36
4.3 The case of GL 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Proof of Wiles’ theorem . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 The case of GL n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.1 Local Langlands correspondence . . . . . . . . . . . . . . . . . . . 40
4.4.2 Automorphic forms on GL n . . . . . . . . . . . . . . . . . . . . . . 42
4.4.3 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.4 A variant, following [CHT08] . . . . . . . . . . . . . . . . . . . . . 43
4.4.5 Newton and Hodge filtration . . . . . . . . . . . . . . . . . . . . . 44
4.4.6 Ordinary and quasi-ordinary representations . . . . . . . . . . . . 44
5 On Local Galois Representations attached to Automorphic Forms 45
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Principal series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.1 Spherical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Ramified principal series case . . . . . . . . . . . . . . . . . . . . . 48
5.3 Steinberg case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5

5.3.1 Unramified twist of Steinberg . . . . . . . . . . . . . . . . . . . . . 49
5.3.2 Ramified twist of Steinberg . . . . . . . . . . . . . . . . . . . . . . 51
5.4 Supercuspidal ⊗ Steinberg . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4.1 m = 2 and n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.4.2 τ unramified supercuspidal of dim m ≥ 2 and n ≥ 2 . . . . . . . . 56
5.4.3 General case: m ≥ 2 and n ≥ 2 . . . . . . . . . . . . . . . . . . . . 58
5.5 General Weil-Deligne representations . . . . . . . . . . . . . . . . . . . . . 62
5.5.1 Sum of twisted Steinberg . . . . . . . . . . . . . . . . . . . . . . . 62
5.5.2 General ordinary case . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.6 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6 2-adic Hida theory and Applications 69
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1.1 Group and sheaf cohomology . . . . . . . . . . . . . . . . . . . . . 70
6.1.2 Hecke action on the cohomology groups . . . . . . . . . . . . . . . 71
6.1.3 Geometry of Riemann surfaces . . . . . . . . . . . . . . . . . . . . 72
6.1.4 Hida’s idempotent operator . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Relations between cohomology groups with coefficients . . . . . . . . . . . 73
6.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4 Control theorem for cohomology . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 Freeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.6 Constant rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.7 Λ-adic Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.8 Control theorem for ordinary Hecke algebras . . . . . . . . . . . . . . . . 89
6.9 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.9.1 2-adic Λ-adic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.9.2 Tame level N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.9.3 Uniqueness result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.10 An application to Galois representations . . . . . . . . . . . . . . . . . . . 94
6.10.1 Buzzard’s result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.10.2 Λ-adic Galois representations . . . . . . . . . . . . . . . . . . . . . 96
6.10.3 Local splitting for Λ-adic forms . . . . . . . . . . . . . . . . . . . . 97
6.10.4 Descending to the classical situation . . . . . . . . . . . . . . . . . 98
A A 2-adic Control Theorem for Modular Curves 99
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.4 Ordinary parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A.5 Iwasawa modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
A.6 Limits of cohomology modules . . . . . . . . . . . . . . . . . . . . . . . . 107
6

This thesis is dedicated to
my parents for their endless Love, Encouragement, and Belief.

ACKNOWLEDGEMENTS
This thesis would not have been possible without the warm encouragement and the guidance
of my supervisor Prof. Eknath Ghate. I owe my deepest gratitude to him for his
patience and care. During all these years, I learnt not only mathematics but also many
other things which made me a better human being.
I take this opportunity to thank all the members of the School of Mathematics for
their patience and help.
I wish to thank Prof. S. Bhattacharya, Prof. N. Nitsure, and Prof. V. Srinivas for the
excellent first year graduate courses. I would also like to thank Prof. S. M. Bhatwadekar,
Prof. R. V. Gurjar, and Prof. D. Prasad for the second year graduate courses.
I wish to thank Prof. N. Nitsure not only for his help at the beginning of my Ph. D.,
but also for keeping an eye on me throughout. Over the years, I gradually understood
Prof. V. Srinivas’s thoughtful comments about life and mathematics. I really appreciate
them now.
I wish to express my deep-felt regards to Prof. D. Prasad for the mathematical
discussions at TIFR and also at Number theory seminars at IIT, Mumbai. I also thank
him for his encouragement and support at various stages.
I would like to thank Prof. R. Sujatha for various discussions at TIFR and also during
workshops at Chennai and Guwahati. I would also like to thank Prof. C. S. Rajan and
Prof. Saradha for discussing mathematics on several occasions.
I was surprised by the amount of energy and enthusiasm of Prof. M. S. Raghunathan
even at this age. I thank him for explaining various mathematical concepts, their origins,
and many more things. I also thank Prof. Ravi Rao for being very co-operative at many
places, and also for providing helpful suggestions whenever needed.
I wish to thank Prof. H. Hida for answering several of my questions patiently at several
stages of my Ph. D. Thanks to Prof. K. Buzzard and Prof. M. Emerton for their helpful
e-mail communications at the beginning of my Ph. D. I would like to thank Prof. U. K.
Anandavardhanan and Prof. R. Raghunathan for making me a part of various Number
theory activities at IIT, Mumbai. Special thanks to Prof. U. K. Anandavardhanan for
being friendly and helpful since the beginning of my Ph. D.
This work was partly completed during a visit to Université Paris-Sud-11, Paris supported
by the ARCUS program (Région Ile-de-France). I am grateful to Prof. L. Clozel
and Prof. J. Tilouine for arranging the visit. I would also like to thank Hausdorff Research
Institute for Mathematics, Bonn, where part of this work was carried out during
the Trimester Program “Algebra and Number Theory”.
I would like to thank Profs. C. Breuil, D. A. Elwood, A. Mézard, M-F. Vignéras, and
J-P. Wintenberger, the organizers of the “Galois Trimester” at IHP, Paris for providing
me a platform to talk to hundreds of mathematicians from all over the world. I benefitted
from the various courses and the conferences held during the “Galois Trimester”.
Words are inadequate in offering my thanks to my teacher Prof. M. Perisastri, whose
influence first made mathematics interesting for me during my undergraduate days.
I take this opportunity to thank all the members of Department of Mathematics and
Statistics, University of Hyderabad. My special thanks to Prof. T. Amaranath, Prof. M.
9

S. Datt, Prof. V. Kannan, Prof. C. Musili (late), Prof. V. Suresh, and Prof. R. Tandon
for their excellent courses at M. Sc., and also for their encouragement in pursuing further
studies.
I have benefitted from Mathematics Training and Talent Search, Advanced Foundational
Schools and Advanced Instructional Schools. I thank all the organizers, especially
Prof. S. Kumaresan, Prof. A. R. Shastri, and Prof. J. K. Verma, for conducting such
perfect programs.
I take this opportunity to thank my M. Sc. batchmates Ayyangar, Damodar, Devakar,
Karthik, Keshav, Pratyusha, Preena, Sanjay, Siva, and Srikanth for their continuous
encouragement. I am also thankful to Prem, Shyam, Ved and Ranjana.
I would like to thank all my friends at TIFR for making my stay enjoyable. I would
specially like to thank Alok, Amala, Amit, Arati, Arijit, Arnab, Chandrasheel and Prachi,
Debargha, Illangovan, Niraj, Ronnie, Sagar, Sandeep, Satadal, Shanta, Shilpa, Somnath,
Souradeep, Sudarshan, Vaibhav, and Vivek from Mathematics, Argha, Aditya, Bhargav,
Manjusha, Nikhil, Partha, Rakesh, Sangeetha, Sashi, and Satej from Physics, Atul, Krishnamohan,
Manoj, and Ram from Chemistry, Aksar, Benny, Harshit, Kalyan, Mallickarjun,
Santanu, and Saswata from Computer Science, Harinath, and Sunilnoothi from Biology.
Each one has some good qualities which are unique to them. I tried to acquire them but
I succeeded only a bit.
I am short of words to express my sincere thanks to Chandrakant, Naren HR, Narendra,
Sarang, and Vijay for our long discussions on various things, and also for the amount
of help that I took from them in, and out of, mathematics.
I would like to thank all the members, particularly Mr. D. B. Sawant and Mr. K.
G. Jayaraj, of the School of Mathematics office, for making all administrative issues so
simple and smooth. Prof. P. A. Gastesi and Mr. V. Nandagopal deserve special thanks
for their help with software related issues.
Above all, I take immense pleasure in thanking my parents, whose support has been
instrumental in overcoming several hurdles in my life, my siblings and their families for
their extended support, and finally to my late grand mother and my late uncle Ch. S. P.
Ranga Rao, who told me about many aspects of life during my childhood. I owe every
bit of my existence to them.
10

Chapter 1
Synopsis
My thesis consists of two parts, one a general result and the other rather specific. The former
deals with the irreducibility of the (p, p)-Galois representations attached to automorphic
representations of GL n (A Q ), for n ≥ 3, and the latter deals with the semisimplicity
of the local Galois representations attached to normalized ordinary cuspidal eigenforms
(n = 2), for the prime p = 2.
The starting point for both parts of my thesis is the following theorem, which is due
to Wiles. Let f = ∑ ∞
n=1 a n(f)q n be a normalized cuspidal eigenform of weight k ≥ 2. Let
K f denote the number field generated by the a n (f)’s. Let G p denote the decomposition
group at ℘, where ℘ denotes the prime induced by a fixed embedding of ¯Q in ¯Q p . There
exists a Galois representation
ρ f,℘ : Gal( ¯Q/Q) → GL 2 (K f,℘ ),
associated to f (and ℘). If f is ordinary at ℘, i.e., a p (f) is a ℘-adic unit, then the
representation ρ f,℘ | Gp , the restriction of ρ f,℘ to G p , is reducible [Wil88]. Moreover, the
powers of the p-adic cyclotomic character χ cyc,p occurring in the semisimplification of
ρ f,℘ | Gp are distinct and related to the weight k of f.
The local representation above is referred to as the (p, p)-Galois representation attached
to f.
1.1 (p, p)-Galois representations attached to automorphic
forms on GL n
In the first part our thesis (cf. [GK1]), we study the local reducibility at p of the p-adic
Galois representation attached to a cuspidal automorphic representation of GL n (A Q ),
assuming that the global representation lives in a strictly compatible system of Galois
representations. In the case that the underlying Weil-Deligne representation is Frobenius
semisimple and indecomposable, we analyze the reducibility completely. We use methods
from p-adic Hodge theory, and work under a transversality assumption on the Hodge and
Newton filtrations in the corresponding filtered module.
Let π be a cuspidal automorphic representation of GL n (A Q ) with infinitesimal character
H consisting of n-distinct integers. The source of (p, p)-Galois representations is
the following conjecture [Tay04, Conj. 3.4] which has been proved by Kottwitz, Clozel,
11

and Harris-Taylor, for certain self-dual cuspidal π’s, and is also true when n = 2 and
π = π(f), for f as above.
Conjecture: Let π be a cuspidal automorphic representation of GL n (A Q ) with infinitesimal
character H consisting of integers. Then there is an irreducible geometric
strongly compatible system of l-adic representations with Hodge-Tate weights H, such
that Local-Global compatibility holds for all primes p.
For every prime p, the conjecture provides a semisimple p-adic Galois representation
ρ π,p : Gal( ¯Q/Q) → GL n ( ¯Q p ). The (p, p)-Galois representation ρ π,p | Gp , the restriction of
ρ π,p to G p , is potentially semistable with Hodge-Tate weights H. By work of Colmez-
Fontaine, we can associate a crystal D(ρ π,p | Gp ) to ρ π,p | Gp . The study of the irreducibility
of the representation ρ π,p | Gp reduces to studying D(ρ π,p | Gp ), because the existence of an
admissible submodule of D(ρ π,p | Gp ) is equivalent to the reducibility of the representation
ρ π,p | Gp .
Let WD(ρ π,p | Gp ) denote the (p, p)-Weil-Deligne representation associated to ρ π,p | Gp .
By the definition of a strictly compatible system, we know WD(ρ π,p | Gp ) from the (p, l)-
Weil-Deligne representation for l ≠ p. Deligne classified all such Frobenius semisimple
(p, l)-Weil-Deligne representations as the direct sum of the indecomposable representations
τ m ⊗ Sp(n), where τ m is an irreducible representation of dimension m of W p , the
Weil group of Q p , and Sp(n) denotes the Steinberg representation of dimension n.
Principal series
Let π be a cuspidal automorphic representation of GL m (A Q ) with infinitesimal character
given by the integers −β 1 > · · · > −β m , i.e., H = {β 1 , . . . , β m }. Suppose π p , the local
component of π at p, is an unramified principal series representation with Satake parameters
α 1 , . . . , α m . In this case, WD(ρ π,p | Gp ) is a direct sum of m unramified characters.
Theorem 1.1.1. If π is ordinary at p, i.e., β i + v p (α i ) = 0 for all i, then ρ π,p | Gp ∼
⎛
α
λ( 1
) · χ −β ⎞
1
p vp(α 1 ) cyc,p ∗ · · · ∗
α
0 λ( 2
) · χ −β 2
p
⎜
vp(α 2 ) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ ,
α
0 0 0 λ( m
)) · χ −βm
p vp(αm) cyc,p
where λ(x) is the unramified character taking arithmetic Frobenius to x. In particular,
the representation ρ π,p | Gp is ordinary.
Indecomposable case
Let π be a cuspidal automorphic representation of GL mn (A Q ) with infinitesimal character
given by the integers −β 1 > · · · > −β mn . Suppose that the Weil-Deligne representation
attached to π p is Frobenius semisimple and indecomposable, i.e.,
WD(ρ π,p | Gp ) ∼ τ m ⊗ Sp(n).
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Twist of Steinberg
When m = 1, we prove:
Theorem 1.1.2. If π is ordinary at p, i.e., β 1 + v p (α) = 0, then the β i are necessarily
consecutive integers, and ρ π,p | Gp ∼
⎛
α
χ 0 · λ( ) · χ −β ⎞
1
p vp(α) cyc,p ∗ · · · ∗
α
0 χ 0 · λ( ) · χ −β 1−1
p
⎜
vp(α) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ ,
α
0 0 0 χ 0 · λ( ) · χ −β 1−(n−1)
p vp(α) cyc,p
where the character τ 1 decomposes as χ 0 · χ ′ , where χ 0 is the ramified part, and χ ′ is an
unramified character mapping arithmetic Frobenius to α.
If π is not ordinary at p, then the local representation ρ π,p | Gp is irreducible.
The last statement generalizes a well-known fact for non-ordinary (p, p)-Galois representations
attached to normalized cuspidal eigenforms (n = 2) at Steinberg primes.
Supercuspidal ⊗ Steinberg
Now we assume that τ m is an irreducible representation corresponding to a supercuspidal
representation of GL m for m ≥ 2. Let E be the field of coefficients of the representation
ρ π,p | Gp and let F be a finite Galois extension of Q p such that τ m | IF = 1. There is an
equivalence of categories between (ϕ, N, F, E)-modules and Weil-Deligne representations.
We write D τm and D Sp(n) for the (ϕ, N, F, E)-modules corresponding to τ m and Sp(n),
respectively. We prove:
Theorem 1.1.3. All the (ϕ, N, F, E)-submodules of D = D τm ⊗ D Sp(n) are of the form
D τm ⊗ D Sp(r) , for some 1 ≤ r ≤ n.
When τ m ∼ Ind Wp
W χ, where χ is a character of W p m pm, we also write down the filtered
(ϕ, N, F, E)-module with underlying Weil-Deligne representation τ m , generalizing results
in [GM09] for m = 2.
Let π be a cuspidal automorphic representation of GL mn (A Q ) with infinitesimal character
consisting of distinct integers {β i,j } i=n,j=m
i=1,j=1
ordered as follows: β i1 ,j 1
> β i2 ,j 2
, if
i 1 > i 2 , or if i 1 = i 2 and j 1 > j 2 . Let t N (D τm ) denote the Newton number of the
(ϕ, N, F, E)-module D τm . Using Theorem 1.1.3, we prove:
Theorem 1.1.4. Suppose that
WD(ρ π,p | Gp ) ∼ τ m ⊗ Sp(n).
If π is ordinary at p, i.e., t N (D τm ) = ∑ m
j=1 β 1,j, then ρ π,p | Gp is reducible, in which case
m = 1, τ 1 is a character, and ρ π,p | Gp is quasi-ordinary as in Theorem 1.1.2.
If π is not ordinary at p, then ρ π,p | Gp is irreducible. In particular, when m ≥ 2, the
representation ρ π,p | Gp is always irreducible.
The theorems above gives complete information about the reducibility of the (p, p)-
Galois representation in the indecomposable case.
13

Decomposable case
Suppose π is a cuspidal automorphic representation of GL N (A Q ) with infinitesimal character
given by the integers −β 1 > · · · > −β N . Suppose that N = ∑ r
i=1 m in i and
WD(ρ π,p | Gp ) ∼ ⊕ r i=1 τ m i
⊗ Sp(n i ). In this case, we can also define p-ordinariness in terms
of the t N (D τi )’s and β i ’s and show that there exists an admissible flag of submodules in
the corresponding (ϕ, N, F, E)-module.
Theorem 1.1.5. If π is ordinary at p, then m i = 1 for all i, the β i occur in r blocks of
consecutive integers, of lengths n i , for 1 ≤ i ≤ r, and
⎛
⎞
ρ n1 ∗ · · · ∗
0 ρ n2 · · · ∗
ρ π,p | Gp ∼ ⎜
⎟
⎝ 0 0 · · · ∗ ⎠ ,
0 0 0 ρ nr
where each ρ ni is an n i -dimensional representation with shape similar to that in Theorem
1.1.2. In particular, ρ π,p | Gp is quasi-ordinary.
In Theorem 1.1.5, we show that our ordinariness condition implies that a particular
complete flag is admissible. We have constructed examples for which not all complete flags
are necessarily admissible, even under our ordinariness assumption. In the decomposable
case, we give an example of a reducible (p, p)-Galois representation such that there is no
complete flag of reducible submodules. Finally, in the non-ordinary decomposable case,
we have constructed examples where the crystal is irreducible.
1.2 2-adic Hida theory and applications
In the second part of our thesis (cf. [GK2], [Kum]), we study the semisimplicity of the
local Galois representations attached to normalized ordinary cuspidal eigenforms (n = 2),
for the prime p = 2.
2-adic Hida theory
We first prove a control theorem for the Hida’s ordinary Λ-adic Hecke algebra for the
prime p = 2 by following the approach of Hida [Hid86a] which deals with primes p ≥ 5.
Let T 2 = Q 2 /Z 2 and Γ r = 1 + 2 r Z 2 of Γ 2 = 1 + 4Z 2 = 〈u〉. For r ≥ 2, we write
V r = H 1 (X r , T 2 ) and W r = H 1 (Y r , T 2 ), where X r denotes the smooth compactification
of the complex manifold Y r = Γ 1 (N2 r )\H, with (2, N) = 1. Let Vr 0 and Wr 0 denote the
ordinary parts of V r and W r , respectively, where the ordinary parts are defined by using
Hida’s idempotent operator. There is a natural action of Γ 0 (N2 r )/Γ 1 (N2 r ) on Vr
0 and
Wr 0 . Let V 0 and W 0 denote the direct limit of Vr 0 and Wr 0 , respectively. Since (Z/N2 r Z) ×
acts on Vr 0 and Wr 0 , Z × 2 acts on V0 and W 0 . Let V 0 and W 0 denote the Pontryagin duals
of V 0 and W 0 , respectively. These are Λ-modules for Λ = Z 2 [[Γ 2 ]].
A first step towards proving a control theorem for the Hida’s ordinary Λ-adic Hecke
algebra is to prove a control theorem for the cohomology modules above.
14

Theorem 1.2.1. 1. For each positive integer r ≥ 2, the restriction morphism of cohomology
groups induces an isomorphism of Vr 0 onto (V 0 ) Γr . The same result also
holds for W 0 .
2. Let N > 1. The modules V 0 and W 0 are free of finite rank over Λ.
The proof of the freeness of V 0 depends heavily on the Z 2 -freeness of the module
e(H 1 (Φ 2 , Z 2 (ω))/H 1 p(Φ 2 , Z 2 (ω))), where Φ 2 = Γ 1 (N)∩Γ 0 (4), and on the following theorem.
Theorem 1.2.2. For each positive integer n ≡ a (mod 2),
rank Z2 h 0 n+2(Φ 2 , Z 2 ) = r(a),
where r(a) is the Z 2 -rank of the Hecke algebra h 0 2 (Φ 2, ω a , Z 2 ), where ω is the mod 4
cyclotomic character, i.e., ω(x) = ±1, if x ≡ ±1 (mod 4).
For p ≥ 5, the theorem above follows from [Hid86b, Thm. 3.1 and Cor. 3.2]. But their
proofs depend on the theory of Katz modular forms, the theory of mod p modular forms,
and some results in these theories only hold for p ≥ 5. For p = 2, we give an alternative
proof of this theorem, which completely avoids both these technicalities.
Let K be a finite extension of Q 2 and O K be the integral closure of Z 2 in K. Write Λ
again for O K [[Γ 2 ]]. Let ι denote the natural inclusion of Γ 2 in Λ. For each weight k ≥ 2,
let h 0 k (Γ 1(N2 ∞ ), O K ) denote the ordinary Λ-adic Hecke algebra lim h 0 ←− k (Γ 1(N2 r ), O K ). For
r≥2
k 1 ≥ k 2 ≥ 2, there exists a surjection
h 0 k 1
(Γ 1 (N2 ∞ ), O K ) ↠ h 0 k 2
(Γ 1 (N2 ∞ ), O K ),
and this is an isomorphism by [Hid88b, Thm. 3.2]. Thus the ordinary Λ-adic Hecke
algebra h 0 k (Γ 1(N2 ∞ ), O K ) is independent of the weight, and we denote it by h 0 (N, O K ).
From Theorem 1.2.1 and 1.2.2, we deduce the following theorem, which is the 2-adic
analogue of the control theorem for Hida’s ordinary Λ-adic Hecke algebra h 0 (N, O K ).
Theorem 1.2.3 (Control theorem). For r ≥ 2, the map
ρ k,ɛ : h 0 (N, O K ) ⊗ Λ Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ ↠ h 0 k (Φ2 r, ɛ, K),
where Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ is identified with K, via u corresponds to u k ɛ(u), is an isomorphism,
where P k,ɛ = (ι(u) − ɛ(u)u k ) is an ideal of Λ and ɛ is a character of Γ/Γ r with values in
O K .
As a consequence of the control theorem, we are able to deduce that in a primitive 2-
ordinary Hida family, either all arithmetic specializations are CM forms or no arithmetic
specialization is a CM form. Hence, we can speak of CM and non-CM primitive 2-ordinary
Hida families.
An application
It is known that if f is ordinary at ℘ and f has complex multiplication (CM), then ρ f,℘ | Gp
splits. Greenberg has asked whether the converse also holds. Serre has proved that the
15

converse holds in the case of elliptic curves, i.e., for an elliptic curve E, the representation
ρ E,℘ splits at p if and only if E has CM.
In [GV04], Ghate and Vatsal have proved the converse for odd primes p and for primitive
p-ordinary Λ-adic forms F, using a result of Buzzard [Buz03] for weight one modular
forms and under some conditions on the residual representation ¯ρ F . As a consequence,
they are able to deduce that all arithmetic specializations of a primitive p-ordinary non-
CM Hida family have non-split local Galois representation, except for a possible finite
set of exceptions. By Hida’s control theorem for odd primes p, it is known that this
exceptional set does not contain any CM forms.
We prove a similar result for the case of p = 2, assuming a relevant result of Buzzard
continues to hold for p = 2 in the residually dihedral setting.
Theorem 1.2.4. Let p = 2. Let F be a primitive p-ordinary Hida family of eigenforms
with the property that
1. ¯ρ F is p-distinguished,
2. ¯ρ F is absolutely irreducible, when restricted to Gal( ¯Q/Q(i)),
3. ¯ρ F (c) ≠ 1 and ¯ρ F is both α-modular and β-modular.
Then,
(a) F is CM if and only if ρ F | Gp
splits.
(b) If F is non-CM, then for all but except possibly finitely many arithmetic specializations
f of F, the representation ρ f,℘ | Gp is non-split. Moreover, the possible exceptions
are necessarily non-CM forms by Theorem 1.2.3.
16

Chapter 2
Introduction
In this thesis, we study the p-adic Galois representations of G p := Gal( ¯Q p /Q p ) attached
to automorphic representation of GL n , using p-adic Hodge theory and Hida theory.
Let f = ∑ ∞
n=1 a n(f)q n be a normalized eigenform (i.e., a primitive form) of weight
k ≥ 2, level N ≥ 1, and nebentypus χ : (Z/NZ) × → C × . Let ℘ be the prime of ¯Q
determined by a fixed embedding of ¯Q into ¯Q p . Let ℘ also denote the induced prime of
K f = Q(a n (f)), the Hecke field of f, and let K f,℘ denote the completion of K f at ℘.
There is a global Galois representation
ρ f,℘ : G Q → GL 2 (K f,℘ )
associated to f (and ℘) by Deligne which has the property that for all primes l ∤ Np,
trace(ρ f,℘ (Frob l )) = a l (f) and det(ρ f,℘ (Frob l )) = χ(l)l k−1 ,
where Frob l denotes the arithmetic Frobenius. It is a well-known result of Ribet that
the global representation ρ f,℘ is irreducible. However, if f is ordinary at ℘, i.e., a p (f)
is a ℘-adic unit, then an important theorem of Wiles says that the corresponding local
representation is reducible. Let χ cyc,p denote the p-adic cyclotomic character.
Theorem 2.0.5 ([Wil88]). Let f be a ℘-ordinary primitive form as above. Then the
restriction of ρ f,℘ to the decomposition subgroup G p is reducible. More precisely, there
exists a basis in which
(
)
χ p · λ(β/p k−1 ) · χ k−1
cyc,p u
ρ f,℘ | Gp ∼
,
0 λ(α)
where χ = χ p χ ′ is the decomposition of χ into its p and prime-to-p-parts, λ(x) : G p →
K × f,℘ is the unramified character which takes arithmetic Frobenius to x, and u : G p → K f,℘
is a continuous function. Here α is (i) the unit root of X 2 − a p (f)X + p k−1 χ(p) if p ∤ N,
(ii) the unit a p (f) if p||N, p ∤ cond(χ) and k = 2, and (iii) the unit a p (f) if p|N,
v p (N) = v p (cond(χ)). In all cases αβ = χ ′ (p)p k−1 .
Moreover, in case (ii), a p (f) is a unit if and only if k = 2, and one can easily show that
ρ f,℘ | Gp is irreducible when k > 2.
Urban has generalized Theorem 2.0.5 to the case of primitive Siegel modular cusprepresentations
of genus 2. For a cuspidal automorphic representation π on GSp 4 (A Q )
17

whose Archimedean component π ∞ belongs to the discrete series with cohomological
weights (a, b; a + b) with a ≥ b ≥ 0, Laumon, Taylor, and Weissauer have defined a
four-dimensional Galois representation
ρ π,p : G Q → GL 4 ( ¯Q p )
with standard properties. For an unramified prime p for π, Tilouine and Urban have
generalized the notion of ordinariness for such primes p in three ways. For example, in
the Borel case, the p-ordinariness of π implies that the valuations of the roots of the Hecke
polynomial of π p are 0, b + 1, a + 2 and a + b + 3. In this case, the restriction of ρ π,p
to the decomposition subgroup G p is upper-triangular. More precisely, there is a basis in
which the local representation ρ π,p | Gp
⎛
⎞
λ(δ/p a+b+3 ) · χ a+b+3
cyc,p ∗ ∗ ∗
0 λ(γ/p a+2 ) · χ a+2
cyc,p ∗ ∗
∼ ⎜
⎝ 0 0 λ(β/p b+1 ) · χ b+1 ⎟
cyc,p ∗ ⎠ ,
0 0 0 λ(α)
where α, β, γ and δ are the roots of the Hecke polynomial of π p with increasing valuations.
In the first part of the thesis (cf. Chapters 4 and 5), motivated by the above two
examples, we define a notion of p-ordinariness for cuspidal automorphic representations
π on GL n (A Q ), for n ≥ 3, and we study the reducibility or the irreducibility of the local
Galois representations attached to π [GK1]. We show that p-ordinariness for π implies that
the associated local Galois representation at p is quasi-ordinary, in the sense of Greenberg
(cf. §4.4.6). In this case we explicitly write down the characters on the diagonal. In
certain cases, we also show that the non-ordinariness of π implies the irreducibility of the
p-adic local Galois representation. These results can be thought of as a generalization of
Wiles’ theorem for GL 2 (cf. Theorem 2.0.5), when we interpret the primitive form f as a
cuspidal automorphic representation of GL 2 (A Q ).
In Chapter 4, after stating the necessary results from p-adic Hodge theory, we reprove
Theorem 2.0.5. The proof serves to illustrate the techniques that will be used in Chapter
5. Let f be a primitive form which is ℘-ordinary. A key ingredient in our proof is that the
representation ρ f,℘ lives in a strictly compatible system of Galois representations (ρ f,λ ),
where λ varies over the primes of K f . By Fontaine theory, one can associate a crystal
to the potentially semistable representation ρ f,℘ | Gp . Using the explicit description of the
crystal (cf. [Bre01], [GM09]) in the Unramified, Steinberg, and ramified principal series
cases, we study the reducibility or the irreducibility of the representation ρ f,℘ | Gp .
In Chapter 5, we prove structure theorems for the local Galois representations attached
to cuspidal automorphic representation of GL n (A Q ).
Let π be a cuspidal automorphic representation of GL n (A Q ) with infinitesimal character
H, where H consists of a set of distinct integers. We assume that the global p-adic
Galois representation ρ π,p attached to π exists, and that it satisfies several natural properties,
e.g., it lives in a strictly compatible system of Galois representations, and satisfies
Local-Global compatibility.
By a result of Flath, π is a restricted tensor product π = ⊗ ′ pπ p (cf. [Bum97, Thm.
3.3.3]) of local automorphic representations. Langlands classified that all irreducible admissible
representations of GL n (Q p ) are isomorphic to Q(∆ 1 , . . . , ∆ r ) for some segments
18

{∆ i } r i=1 , such that for i < j, ∆ i does not precede ∆ j (cf. Theorem 4.4.1). There exists
a bijection between isomorphism classes of irreducible admissible representations of
GL n (Q p ) and isomorphism classes of admissible n-dimensional representations, over ¯Q l
for l ≠ p, of W p, ′ the Weil-Deligne group of Q p (cf. the discussion after Theorem 4.4.2).
Deligne classified all such admissible n-dimensional representations of W p, ′ over an
algebraically closed fields of characterstic 0, as the direct sum of the indecomposable
representations τ m ⊗ Sp(n), where τ m is an irreducible representation of dimension m of
W p , the Weil group of Q p , and Sp(r) denotes the Steinberg representation of dimension
r.
Let WD(ρ π,p | Gp ) denote the (p, p)-Weil-Deligne representation associated to ρ π,p | Gp .
By the definition of a strictly compatible system, we know WD(ρ π,p | Gp ) from the (p, l)-
Weil-Deligne representation for l ≠ p, and which is determined by the local automorphic
representation of π at p.
The notion of p-ordinariness is defined by using p-adic valuation v p of certain parameters
coming from the local automorphic representation π at p. The parameters vary in
different cases, but can be made completely precise.
For instance, suppose that the local automorphic representation π p is an unramified
representation of GL m (Q p ) (in the classical setting, we are in this case if p ∤ N), i.e.,
π p = Q(χ 1 , . . . , χ m ), for some unramified characters χ i of Q × p . We can parametrize
the isomorphism class of this representation by the Satake parameters α 1 , . . . , α m , for
α i = χ i (ω), where ω is a uniformizer for Q p . In this case, WD(ρ π,p | Gp ) is a direct sum
of unramified characters χ i ’s. The local representation ρ π,p | Gp is crystalline with Hodge-
Tate weights H. Let D be the corresponding filtered ϕ-module. Let the jumps in the
filtration on D be β 1 < · · · < β m (so that H = {−β 1 , . . . , −β m }). In this case, we say
that the cuspidal automorphic representation π is p-ordinary if
for all i = 1, . . . , m.
β i + v p (α i ) = 0,
Theorem 2.0.6 (Spherical case). Let π be a cuspidal automorphic representation of
GL m (A Q ) with infinitesimal character given by the integers −β 1 > · · · > −β m and such
that π p is in the unramified principal series with Satake parameters α 1 , . . . , α m . If π is
p-ordinary, then ρ π,p | Gp ∼
⎛
α
λ( 1
) · χ −β ⎞
1
p vp(α 1 ) cyc,p ∗ · · · ∗
α
0 λ( 2
) · χ −β 2
p
⎜
vp(α 2 ) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ .
α
0 0 0 λ( n
) · χ −βm
p vp(αm) cyc,p
In particular, ρ π,p | Gp
is ordinary.
A similar result was also obtained by D. Geraghty in the course of proving modularity
lifting theorems for GL n . However, the main point of our thesis is to treat other cases (in
the classical language, the primes of bad reduction).
In the non-principal series case, we no longer have the Satake parameters of π p at our
disposal. However, we can replace these numbers by the corresponding eigenvalues of l-
adic Frobenius in the l-adic Weil-Deligne representation corresponding to π p , for l ≠ p, or
19

equivalently, by using the properties of strictly compatible systems, with the eigenvalues
of crystalline Frobenius on the filtered module attached to π p (as in [GM09] for n = 2).
To keep the analysis of the structure of the local Galois representation ρ π,p | Gp within
reasonable limits in this thesis, we shall assume that the Newton filtration is in general
position with respect to the Hodge filtration on the associated crystal.
Now suppose π p is a twist of the Steinberg representation, i.e., π p = Q(∆) for a
segment ∆ = [σ, σ(n − 1)], where σ is a supercuspidal representation of GL m (Q p ) (in the
classical setting, we are in this case if v p (N) = 1 and v p (cond(χ)) = 0). We know that
the corresponding Frobenius semisimplification of the Weil-Deligne representation is of
the form τ m ⊗ Sp(n). In this case, we not only show that the p-ordinariness of π implies
quasi-ordinariness of the representation ρ π,p | Gp but also show that the non-ordinariness
of π implies the representation ρ π,p | Gp is irreducible, unlike in the principal series case.
When m = 1, we prove:
Theorem 2.0.7 (Twist of Steinberg). Let π be a cuspidal automorphic representation of
GL n (A Q ) with infinitesimal character given by the integers −β 1 > · · · > −β n . Suppose
π p is a twist of the Steinberg representation, and WD(ρ π,p | Gp ) is Frobenius semisimple,
hence
WD(ρ π,p | Gp ) ∼ τ 1 ⊗ Sp(n).
If π is p-ordinary, i.e., β 1 + v p (α) = 0, then the β i are necessarily consecutive integers,
and ρ π,p | Gp ∼
⎛
α
χ 0 · λ( ) · χ −β ⎞
1
p vp(α) cyc,p ∗ · · · ∗
α
0 χ 0 · λ( ) · χ −β 1−1
p
⎜
vp(α) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ ,
α
0 0 0 χ 0 · λ( ) · χ −β 1−(n−1)
p vp(α) cyc,p
where the character τ 1 decomposes as χ 0 · χ ′ , where χ 0 is the ramified part, and χ ′ is an
unramified character mapping arithmetic Frobenius to α.
If π is not ordinary at p, then the local representation ρ π,p | Gp is irreducible.
The last statement generalizes a well-known fact for non-ordinary (p, p)-Galois representations
attached to normalized cuspidal eigenforms (n = 2) at Steinberg primes.
We now turn to the case where m ≥ 2. Recall that π p = Q(∆) for some segment
∆ = [σ, σ(n − 1)], where σ is supercuspidal representation of GL m (Q p ). Suppose the
infinitesimal character of π consists of distinct integers {β i,j } i=n,j=m
i=1,j=1
ordered as follows:
β i1 ,j 1
> β i2 ,j 2
, if i 1 > i 2 , or if i 1 = i 2 and j 1 > j 2 . Before defining the p-ordinariness in
this case, we slightly deviate and study the structure of the associated crystal.
Let E be the field of coefficients of the representation ρ π,p | Gp and let F be a finite
Galois extension of Q p such that τ m | IF = 1. By [BS07, Prop. 4.1], there is an equivalence
of categories between (ϕ, N, F, E)-modules and Weil-Deligne representations. We write
D τm and D Sp(n) for the (ϕ, N, F, E)-modules corresponding to τ m and Sp(n), respectively.
We prove:
Theorem 2.0.8. All the (ϕ, N, F, E)-submodules of D = D τm ⊗ D Sp(n) are of the form
D τm ⊗ D Sp(r) , for some 1 ≤ r ≤ n.
20

When τ m ∼ Ind Wp
W χ, where χ is a character of W p m pm, we also write down the filtered
(ϕ, N, F, E)-module with underlying Weil-Deligne representation τ m , generalizing results
in [GM09] for m = 2.
Let t N (D τm ) denote the Newton number of the (ϕ, N, F, E)-module D τm . Using Theorem
2.0.8, we prove:
Theorem 2.0.9 (Indecomposable case). Suppose that
WD(ρ π,p | Gp ) ∼ τ m ⊗ Sp(n).
If π is ordinary at p, i.e., t N (D τm ) = ∑ m
j=1 β 1,j, then ρ π,p | Gp is reducible, in which case
m = 1, τ 1 is a character, and ρ π,p | Gp is quasi-ordinary as in Theorem 2.0.7.
If π is not ordinary at p, then ρ π,p | Gp is irreducible. In particular, when m ≥ 2, the
representation ρ π,p | Gp is always irreducible.
The theorems above give complete information about the reducibility of the (p, p)-
Galois representation in the indecomposable case.
Now suppose π is a cuspidal automorphic representation of GL N (A Q ) with infinitesimal
character given by the integers −β 1 > · · · > −β N , and π p = Q(∆ 1 , . . . , ∆ r ) for some
segments ∆ i = [σ i , σ i (n i −1)] (upto permutation) such that for i < j, ∆ i does not precede
∆ j , where σ i is a supercuspidal representation of GL mi (Q p ). Clearly, N = ∑ r
i=1 m in i and
WD(ρ π,p | Gp ) ∼ ⊕ r i=1 τ m i
⊗ Sp(n i ). In this case, we can also define p-ordinariness in terms
of the t N (D τi )’s and β i ’s and show that there exists an admissible flag of submodules in
the corresponding (ϕ, N, F, E)-module.
Theorem 2.0.10. If π is ordinary at p, then m i = 1 for all i, the β i occur in r blocks of
consecutive integers, of lengths n i , for 1 ≤ i ≤ r, and
⎛
⎞
ρ n1 ∗ · · · ∗
0 ρ n2 · · · ∗
ρ π,p | Gp ∼ ⎜
⎟
⎝ 0 0 · · · ∗ ⎠ ,
0 0 0 ρ nr
where each ρ ni is an n i -dimensional representation with shape similar to that in Theorem
2.0.7. In particular, ρ π,p | Gp is quasi-ordinary.
In Theorem 2.0.10 we show that our ordinariness condition implies that a particular
complete flag is admissible. In §5.6, we have constructed examples for which not all
complete flags are necessarily admissible, even under our ordinariness assumption. In
the decomposable case, we give an example of a reducible (p, p)-Galois representation
such that there is no complete flag of reducible submodules. Finally, in the non-ordinary
decomposable case, we have constructed examples where the crystal is irreducible.
This gives an overview of all the results in the first part of the thesis.
In the second part of the thesis (cf. Chapter 6 and Appendix A), we show that
some of the basic results of Hida theory in the literature stated for odd primes p (and
originally for p ≥ 5 in [Hid86b], [Hid86a]) remain valid for the prime p = 2. We prove
a 2-adic analogue of the control theorem for the Hida’s Hecke algebra [GK2]. As an
application, we study the semisimplicity of the local Galois representations attached to
ordinary primitive forms (n = 2), for the prime p = 2.
21

In Chapter 6, we start by proving a control theorem for cohomology groups. To explain
briefly, for every positive integer r ≥ 2, we write V r = H 1 (X r , T 2 ), W r = H 1 (Y r , T 2 ), where
T 2 = Q 2 /Z 2 . The Hecke algebra h 2 (Γ 1 (N2 r ), Z 2 ) acts on V r and therefore V 0 r , the ordinary
part of V r , is a module over h 0 2 (Γ 1(N2 r ), Z 2 ), the ordinary part of h 2 (Γ 1 (N2 r ), Z 2 ). There
is also an action of Γ 0 (N2 r )/Γ 1 (N2 r ) on V 0 r and W 0 r . Let V 0 denote the direct limit of V 0 r ,
for r ≥ 2, and similarly for W 0 . One sees that V 0 and W 0 become continuous modules
(if we equip them with the discrete topology) over the Iwasawa algebra Λ = Z 2 [[Γ 2 ]],
where Γ r = 1 + 2 r Z 2 , and Γ = Γ 2 = 〈u〉. Let V 0 and W 0 denote the Pontryagin duals
of V 0 and W 0 , respectively. The modules V 0 and W 0 are modules over Λ. We prove (cf.
Theorem 6.3.1)
Theorem 2.0.11. Let p = 2. We have:
1. For each positive integer r ≥ 2, the restriction morphism of cohomology groups
induces an isomorphism of V 0 r onto (V 0 ) Γr . The same result also holds for W 0 .
2. Let N > 1. The modules V 0 and W 0 are free modules of finite rank over Λ.
A proof of the freeness of W 0 forms the content of Appendix A. For the other statements,
especially the freeness of V 0 , by closely following the proof for odd primes, one
realizes that one needs certain results from the theory of Katz modular forms and the
theory of mod p modular forms. Since these results are not known for p = 2, we prove
the theorem above with other ingredients (cf. §6.6).
Let K be a finite extension of Q 2 and let O K denote the integral closure of Z 2 in
K. By [Hid88b, Thm. 3.2], we know that the ordinary Λ-adic Hecke algebra is defined
independently of the weight and we denote this Hecke algebra by h 0 (N, O K ). In §6.8, we
prove a 2-adic version of the Hida’s control theorem for h 0 (N, O K ).
Theorem 2.0.12 (Control theorem). Let ɛ be a character of Γ/Γ r with values in O K ,
and let P k,ɛ denote the ideal (ι(u) − ɛ(u)u k ) of Λ, where ι is an inclusion of Γ in Λ. For
r ≥ 2, the map
ρ k,ɛ : h 0 (N, O K ) ⊗ Λ Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ ↠ h 0 k (Φ2 r, ɛ, K),
where Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ is identified with K, via u corresponds to u k ɛ(u), is an isomorphism.
As a consequence, we deduce a uniqueness theorem (cf. Theorem 6.9.10) for 2-
stabilized ordinary primitive forms in Hida families. Namely, we show that such forms
live in unique primitive 2-ordinary cuspidal families, up to Galois conjugacy, a well-known
result when p is odd. As an application of this result we show that the notion of CM-ness
is pure with respect to 2-adic families, i.e., we show that for a primitive 2-ordinary Λ-adic
cuspidal eigenform F, either all arithmetic specializations are CM forms or no arithmetic
specialization is a CM form (cf. Proposition 6.9.11). In view of this result, we may and
do speak of CM and non-CM 2-adic Hida families.
Recall that f is a primitive ℘-ordinary form of weight at least two. Furthermore, if
f has complex multiplication (CM), then ρ f,℘ | Gp splits at p. R. Greenberg has asked
whether the converse also holds.
In the case of mod p cusp forms, there is an (almost) complete answer to the above
question due to Gross [Gro90] and Coleman-Voloch. Let g = ∑ a n q n be a p-ordinary mod
22

p cuspidal eigenform of Serre weight 2 ≤ k ≤ p, level N with (p, N) = 1, and nebentypus ψ.
Deligne constructed a Galois representation ρ g associated to g with standard properties.
It is a well-known result of Gross and Coleman-Voloch, that ρ g is tamely ramified at
p if and only if g has a companion form, when k ≠ 2 or k ≠ p. If the form g is p-
distinguished, then ρ g is tamely ramified if and only if ρ g splits at p (cf. [Gha05]). Hence,
if g is p-distinguished, then ρ g splits at p if and only if g has a companion form.
If p is prime to N, then it is a theorem of Serre and Tate that if a modular form f
corresponds to an elliptic curve E defined over Q, then ρ f,℘ = ρ E,p splits at p if and only
if E, and therefore f, has CM.
In [GV04], Ghate and Vatsal have proved the converse for odd primes p and for primitive
p-ordinary Λ-adic forms F, using a result of Buzzard [Buz03] for weight one modular
forms and under some conditions on the residual representation ¯ρ F . As a consequence,
they are able to deduce that all arithmetic specializations of a primitive p-ordinary non-
CM Hida family have non-split local Galois representation, except for a possible finite
set of exceptions. By Hida’s control theorem for odd primes p, it is known that this
exceptional set does not contain any CM forms.
Finally, we study the semisimplicity of the local Galois representations and prove a
similar result for the case of p = 2, assuming a result of Buzzard (cf. Theorem 6.10.2)
continues to hold for p = 2 in the residually dihedral setting. We prove (cf. Theorem 6.10.4
and Theorem 6.10.5):
Theorem 2.0.13. Let p = 2. Let F be a primitive p-ordinary Hida family of eigenforms
with the property that
1. ¯ρ F is p-distinguished,
2. ¯ρ F is absolutely irreducible, when restricted to Gal( ¯Q/Q(i)),
3. ¯ρ F (c) ≠ 1 and ¯ρ F is both α-modular and β-modular.
Then,
(a) F is CM if and only if ρ F | Gp
splits.
(b) If F is non-CM, then for all but except possibly finitely many arithmetic specializations
f of F, the representation ρ f,℘ | Gp is non-split. Moreover, the possible exceptions
are necessarily non-CM forms by Theorem 2.0.12.
23

Chapter 3
Modular forms and Hecke algebras
In this chapter, first we recall the definitions of modular forms and cusp forms of weight
k for congruence subgroups. Then we define the abstract Hecke operators for arbitrary
congruence subgroups. Finally, we recall the concept of oldforms, newforms and the
notion of complex multiplication (CM). We closely follow the exposition in [DS05].
3.1 Modular forms
Each element of the modular group SL 2 (Z) can be viewed as an automorphism of the
Riemann sphere Ĉ = C ∪ {∞} by the fractional linear transformation:
( )
γ(τ) = aτ + b
cτ + d , for γ = a b
and τ ∈
c d
Ĉ.
For every N ∈ N, we define
{(
a
Γ 0 (N) :=
c
{(
a
Γ 1 (N) :=
c
{(
a
Γ(N) :=
c
)
}
b
∈ SL 2 (Z) | c ≡ 0 (mod N) ,
d
)
}
b
∈ Γ 0 (N) | a ≡ 1 (mod N) ,
d
)
}
b
∈ Γ 1 (N) | b ≡ 0 (mod N) .
d
Definition 3.1.1. A subgroup Γ of SL 2 (Z) is a congruence subgroup, if Γ(N) ⊆ Γ for
some N ∈ N, in which case Γ is a congruence subgroup of level N.
Let H denote the upper half plane, and let GL + 2 (R) denote the set of matrices in
GL 2 (R) with positive determinant.
For γ = ( )
a b
c d ∈ GL
+
2 (R) and for any integer k, define the weight-k operator [γ] k on
functions f : H → C by
(f[γ] k )(τ) = (det(τ)) k−1 (cτ + d) −k f(γ(τ)), for τ ∈ H.
Since the term cτ + d is never zero or infinity, if f is meromorphic then f[γ] k is also
meromorphic and has the same zeros and poles as f.
25

Definition 3.1.2. Let Γ be a congruence subgroup of SL 2 (Z) and let k be an integer. A
meromorphic function f : H → C is a weakly modular of weight k with respect to Γ, or
simply weakly modular with respect to Γ, if for all γ ∈ Γ,
f[γ] k = f.
To define modular forms for congruence subgroups Γ, we need to make sense of the
“holomorphy”condition at ∞.
Let D = {q ∈ C : |q| < 1} be the open complex unit disk, let D ′ = D − {0}. Each
congruence subgroup Γ of SL 2 (Z) contains a translation matrix of the form ( )
1 h
0 1 : τ ↦→
τ + h for some minimal h ∈ N. Every function f : H → C that is weakly modular
with respect to Γ is hZ-periodic and thus has a corresponding function g : D ′ → C, i.e.,
f(τ) = g(exp( 2πiτ
h
)). If f is also holomorphic on H, then g is holomorphic on D′ and so
it has a Laurent series expansion at q = 0. Define such f to be holomorphic at ∞, if g
extends holomorphically to D. Thus f has a Fourier expansion
f(τ) =
∞∑
a n (f) exp(2πiτn/h).
n=0
A modular form with respect to a congruence subgroup Γ should be holomorphic not
only at ∞ but also at some other points, namely the cusps.
The group SL 2 (Z) acts on Q ∪ {∞}. We define the cusps of Γ to be the Γ-equivalence
classes of Q ∪ {∞}. Since Γ is of finite index in SL 2 (Z), we see that the number of cusps
is finite. Writing any s ∈ Q ∪ {∞} as s = α(∞), holomorphy at s is defined in terms
of holomorphy at ∞ via the [α] k operator. Since f[α] k is holomorphic on H and weakly
modular with respect to α −1 Γα, the notion of its holomorphy at ∞ makes sense.
Definition 3.1.3. Let Γ be a congruence subgroup of SL 2 (Z) and let k be an integer. A
function f : H → C is a modular form of weight k with respect to Γ, if
1. f is holomorphic, and weakly modular with respect to Γ,
2. f[α] k is holomorphic at ∞ for all α ∈ SL 2 (Z).
If in addition, a 0 (f[α] k ) = 0 in the Fourier expansion of f[α] k for all α ∈ SL 2 (Z), then f
is a cusp form of weight k with respect to Γ. The modular forms and cusp forms of weight
k with respect to Γ are denoted by M k (Γ) and S k (Γ), respectively.
Let N be a natural number. For each Dirichlet character χ modulo N, define the
χ-eigenspace of M k (Γ 1 (N)) and S k (Γ 1 (N)) as
M k (N, χ) := {f ∈ M k (Γ 1 (N)) | f[γ] k = χ(d γ )f, for all γ ∈ Γ 0 (N)} , and
S k (N, χ) := M k (N, χ) ∩ S k (Γ 1 (N)).
(Here d γ denotes the lower right entry of γ.) In particular, the eigenspace M k (N, 1) is
M k (Γ 0 (N)). The vector spaces M k (Γ 1 (N)) and S k (Γ 1 (N)) decomposes as the direct sum
of the eigenspaces, i.e.,
M k (Γ 1 (N)) = ⊕ χ M k (N, χ) and S k (Γ 1 (N)) = ⊕ χ S k (N, χ).
26

3.2 Hecke algebras
We now briefly recall the definition of the Hecke operators, which play an important role
in the study of modular forms, and cusp forms. For more details, refer to [Miy89, §2.8]
For any congruence subgroup Γ of SL 2 (Z), we put
˜Γ = {g ∈ GL + 2 (Q) | gΓg−1 and Γ are commensurable}.
Let Γ 1 and Γ 2 be congruence subgroups of SL 2 (Z). It is easy to see that ˜Γ 1 = ˜Γ 2 . Let ∆
be subsemigroup of ˜Γ 1 . For any α ∈ ∆, we set
Γ 1 αΓ 2 = {γ 1 αγ 2 | γ 1 ∈ Γ 1 , γ 2 ∈ Γ 2 }
is a double coset in GL + 2 (Q). Using these double cosets, we define Hecke operators and
they transform modular forms with respect to Γ 1 into modular forms with respect to Γ 2 .
Definition 3.2.1. For congruence subgroups Γ 1 and Γ 2 of SL 2 (Z) and α ∈ ∆, the weightk
operator [Γ 1 αΓ 2 ] k takes functions f ∈ M k (Γ 1 ) to
f[Γ 1 αΓ 2 ] k = ∑ j
f[β j ] k ∈ M k (Γ 2 )
where {β j } are orbit representatives, i.e., Γ 1 αΓ 2 = ∪ j Γ 1 β j is a disjoint union.
By [Miy89, Thm. 2.8.1], the weight-k operator is well-defined and it maps S k (Γ 1 ) into
S k (Γ 2 ). Again by the same theorem, the weight-k operator maps the space of modular
forms M k (Γ 1 , χ) (resp., S k (Γ 1 , χ)) to M k (Γ 2 , χ) (resp., S k (Γ 2 , χ)).
3.2.1 Hecke operators for Γ 1 (N)
The group Γ 0 (N) acts on M k (Γ 1 (N)), and it induces an action of the quotient (Z/NZ) ∗ .
The action of α = ( a b
c d
)
, determined by d (mod N) and denoted 〈d〉, is
〈d〉 : M k (Γ 1 (N)) → M k (Γ 1 (N))
given by 〈d〉f = f[α] k for any α ∈ Γ 0 (N) such that d α ≡ d (mod N). This Hecke operator
is called as diamond operator. By definition, we see that
M k (N, χ) = {f ∈ M k (Γ 1 (N)) | 〈d〉f = χ(d)f for all d ∈ (Z/NZ) ∗ }.
Since the weight-k operator [ ] k preserves the space of cusp forms, we see that the diamond
operator preserves the spaces of cuspforms S k (Γ 1 (N)).
The second type of Hecke operator is also a weight-k operator [Γ 1 (N)αΓ 1 (N)] k , but
now α = ( )
1 0
0 p , for a prime p. This operator is denoted Tp . For primes p dividing N,
sometimes we write U p for T p . Thus
is given by
T p : M k (Γ 1 (N)) → M k (Γ 1 (N)),
T p (f) = f[Γ 1 (N) ( 1 0
0 p
)
Γ1 (N)] k .
Using a left coset decomposition of Γ 1 (N) ( )
1 0
0 p Γ1 (N), we can explicitly write down the
action of [Γ 1 (N) ( )
1 0
0 p Γ1 (N)] k on modular forms.
27

Proposition 3.2.2. Let Γ 1 = Γ 2 = Γ 1 (N), and let α = ( )
1 0
0 p where p is a prime. The
operator T p = [Γ 1 (N) ( )
1 0
0 p Γ1 (N)] k on M k (Γ 1 (N)) is given by
⎧
∑p−1
f[ ( 1 j
)
0 p ]k if p | N,
⎪⎨
j=0
T p (f) =
∑p−1
⎪⎩ f[ ( 1 j
)
0 p ]k + f[ ( m n
)( p 0
)
N p 0 1 ]k if p ∤ N, where mp − nN = 1.
j=0
Letting Γ 1 = Γ 2 = Γ 0 (N) instead and keeping same α as above gives the same orbit
representatives for Γ 1 αΓ 2 , but in this case ( m n
)( p 0
) (
N p 0 1 can be replaced by p 0
)
0 1 .
Using the proposition above, we can describe the effect of T p on Fourier coefficients
as follows: If f ∈ M k (N, χ) then also T p (f) ∈ M k (N, χ) and
a n (T p f) = a np (f) + χ(p)p k−1 a n/p (f) for f ∈ M k (N, χ),
where a n/p (f) = 0, if p ∤ n. Moreover, the action of T p preserves the subspaces S k (Γ 1 (N))
and S k (N, χ).
So far the Hecke operators 〈d〉 and T p are defined for d ∈ (Z/NZ) ∗ and p prime. Now
we extend the definitions to n and T n for all n ∈ N. For n ∈ N with (n, N) = 1, 〈n〉 is
determined by n (mod N). For n ∈ N with (n, N) > 1, define 〈n〉 = 0, the zero operator.
To define T n , set T 1 = 1 (the identity operator); T p is already defined for primes p.
For prime powers, define inductively
T p r = T p T p r−1 − p k−1 〈p〉T p r−2, for r ≥ 2.
Extend the definition multiplicatively to T n for all n,
T n = ∏ i
T p e i
where n = ∏ i
p e i
.
Similar to the above, we can explicitly write the Fourier coefficients of T n (f), for any
f ∈ M k (Γ 1 (N)). If f ∈ M k (Γ 1 (N)), then a m (T n f) = ∑ d|(m,n) dk−1 a mn/d 2(〈d〉f). In
particular, if f ∈ M k (N, χ), then a m (T n f) = ∑ d|(m,n) χ(d)dk−1 a mn/d 2(f).
For N ∈ N, we let h k (Γ 1 (N), C) denote the subring of End C (S k (Γ 1 (N))) generated over
C by the Hecke operators T p for all primes p and the operators 〈d〉 acting on S k (Γ 1 (N)).
Each element f in S k (Γ 1 (N)) has the Fourier expansion f(z) = ∑ n a n(f)q n , for a n (f) ∈
C. Hence, we may embed S k (Γ 1 (N)) into the power series C[[q]]. One may then give
a rational structure on S k (Γ 1 (N)) by defining the A-rational subspace S k (Γ 1 (N), A) for
each subalgebra A of C or C p , by S k (Γ 1 (N), A) := S k (Γ 1 (N)) ∩ A[[q]]. Similar to above,
one can define the Hecke algebra over Z, and hence over A. We denote this Hecke algebra
by h k (Γ 1 (N), A).
3.2.2 Oldforms and Newforms
There is a way to move between levels is to observe that, if M|N then S k (Γ 1 (M)) ⊆
S k (Γ 1 (N)). Another way to embed S k (Γ 1 (M)) into S k (Γ 1 (N)) is by composing with the
multiply-by-d map, where d is any factor of N/M. For any such d, let α d = ( d 0
0 1
)
so that
(f[α d ] k )(τ) = d k−1 f(dτ)
28

for f : H → C. The injective linear map [α d ] k takes S k (Γ 1 (M)) to S k (Γ 1 (N)), lifting the
level from M to N.
Definition 3.2.3. For each divisor d of N, let i d be the map
i d : (S k (Γ 1 (Nd −1 ))) 2 → S k (Γ 1 (N))
given by (f, g) → f + g[α d ] k . The subspace of oldforms of level N is given by
S k (Γ 1 (N)) old = ∑ p|N
i p ((S k (Γ 1 (Np −1 ))) 2
and the subspace of newforms of level N is the orthogonal complement with respect to
the Petersson inner product,
S k (Γ 1 (N)) new = (S k (Γ 1 (N)) old ) ⊥ .
For all n ∈ N, the Hecke operators T n and 〈n〉 preserve the space of oldforms and
newforms. Moreover, the spaces S k (Γ 1 (N)) old and S k (Γ 1 (N)) new have orthogonal bases
of eigenforms for the Hecke operators away from the level, {T n , 〈n〉 : (n, N) = 1}.
Definition 3.2.4. A nonzero modular form f ∈ M k (Γ 1 (N)) that is an eigenform for the
Hecke operators T n and 〈n〉 for all n ∈ N is a Hecke eigenform or simply an eigenform.
The eigenform f(τ) = ∑ ∞
n=0 a n(f)q n is normalized, when a 1 (f) = 1. A newform is a
normalized eigenform in S k (Γ 1 (N)) new .
Theorem 3.2.5 ([DS05]). Let f ∈ S k (Γ 1 (N)) new be a nonzero eigenform for the Hecke
operators T n and 〈n〉 for all n with (n, N) = 1. Then f is an eigenform.
3.2.3 Complex multiplication
We briefly recall the definition of a newform to have complex multiplication. For more
details, refer to [Rib77, §3].
Let f ∈ S k (N, χ) be a newform and let ϕ be a Dirichlet character. We let f ⊗ ϕ be
f ⊗ ϕ := ∑ n
ϕ(n)a n (f)q n .
By [Shi71, Prop. 3.64], f ⊗ ϕ is a modular form of weight k on Γ 1 (ND 2 ) of nebentypus
χϕ 2 , if ϕ is defined mod D. We have (f ⊗ ϕ)| Tp = ϕ(p)a p (f)(f ⊗ ϕ) for p ∤ ND.
Definition 3.2.6. Suppose ϕ ≠ 1. The form f has complex multiplication (or CM for
short) by ϕ, if ϕ(p)a p (f) = a p (f) for all primes p in a set of primes of density 1.
Remark 3.2.7. If f has CM by ϕ, then we have χϕ 2 = χ, and ϕ(p)a p (f) = a p (f) for
all p ∤ DN. The first equation implies that ϕ is a quadratic character. If the kernel of ϕ
defines a imaginary quadratic field F , we say that f has CM by F .
29

Chapter 4
Galois Representations and
Automorphic Forms on GL n
We start the chapter by stating the results (cf. §4.1) which motivated the first part of
the thesis. Then we recall some definitions and some facts from p-adic Hodge theory (cf.
§4.2). In §4.3, we reprove Theorem 4.1.1, using the work of Colmez-Fontaine [CF00], and
it serves to illustrate the techniques that will be used. In §4.4, we recall some generalities
about Galois representations associated to automorphic representation of GL n (A Q ).
4.1 Motivation
For a global or a local field F , we set G F = Gal( ¯F /F ). For simplicity, we write G p for
G Qp . Let ℘ be the prime of ¯Q determined by a fixed embedding of ¯Q into ¯Q p .
Let f = ∑ ∞
n=1 a n(f)q n be a normalized eigenform (i.e., primitive form) of weight
k ≥ 2, level Np r ≥ 1, with (N, p) = 1, and nebentypus χ. Let ℘ also denote the induced
prime of K f = Q(a n (f)), the Hecke field of f, and let K f,℘ denote the completion of K f
at ℘. There is a global Galois representation
ρ f,℘ : G Q → GL 2 (K f,℘ ) (4.1.1)
associated to f (and ℘) by Deligne which has the property that for all primes l ∤ Np,
trace(ρ f,℘ (Frob l )) = a l (f) and det(ρ f,℘ (Frob l )) = χ(l)l k−1 .
Thus det(ρ f,℘ ) = χχ k−1
cyc,p, where χ cyc,p is the p-adic cyclotomic character.
It is a well-known result of Ribet that the global representation ρ f,℘ is irreducible.
However, if f is ordinary at ℘, i.e., a p (f) is a ℘-adic unit, then an important theorem of
Wiles, valid more generally for Hilbert modular forms, says that the corresponding local
representation ρ f,℘ restricted to G p is reducible.
Theorem 4.1.1 ([Wil88]). Let f be a ℘-ordinary primitive form as above. Then the
restriction of ρ f,℘ to the decomposition subgroup G p is reducible. More precisely, there
exists a basis in which
(
)
χ p · λ(β/p k−1 ) · χ k−1
cyc,p u
ρ f,℘ | Gp ∼
, (4.1.2)
0 λ(α)
31

where χ = χ p χ ′ is the decomposition of χ into its p and prime-to-p-parts, λ(x) : G p →
K × f,℘ is the unramified character which takes arithmetic Frobenius to x, and u : G p → K f,℘
is a continuous function. Here α is (i) the unit root of X 2 − a p (f)X + p k−1 χ(p) if p ∤ N,
(ii) the unit a p (f) if p||N, p ∤ cond(χ) and k = 2, and (iii) the unit a p (f) if p|N,
v p (N) = v p (cond(χ)). In all cases αβ = χ ′ (p)p k−1 .
Moreover, in case (ii), a p (f) is a unit if and only if k = 2, and one can easily show that
ρ f,℘ | Gp is irreducible when k > 2.
Urban has generalized Theorem 4.1.1 to the case of primitive Siegel modular cusprepresentations
of genus 2. We briefly recall this result here. Let π be a cuspidal automorphic
representation of GSp 4 (A Q ) whose Archimedean component π ∞ belongs to the discrete
series, with cohomological weights (a, b; a + b) with a ≥ b ≥ 0. For each prime p, Laumon,
Taylor and Weissauer have defined a four-dimensional Galois representation
ρ π,p : G Q → GL 4 ( ¯Q p )
with standard properties. Let p be an unramified prime for π. Then Tilouine and Urban
have generalized the notion of ordinariness for such primes p in three ways to what they
call Borel ordinary, Siegel ordinary, and Klingen ordinary (these terms come from the
underlying parabolic subgroups of GSp 4 (A Q )). In the Borel case, the p-ordinariness of π
implies that the Hecke polynomial of π p , namely
(X − α)(X − β)(X − γ)(X − δ),
has the property that the p-adic valuations of α, β, γ and δ are 0, b+1, a+2 and a+b+3,
respectively.
Theorem 4.1.2 ([Urb05], [TU99]). Say π is a Borel p-ordinary automorphic cusp representation
for GSp 4 (A Q ) which is stable at ∞ with cohomological weights (a, b; a + b).
Then the restriction of ρ π,p to the decomposition subgroup G p is upper-triangular. More
precisely, there is a basis in which ρ π,p | Gp
⎛
⎞
λ(δ/p a+b+3 ) · χ a+b+3
cyc,p ∗ ∗ ∗
0 λ(γ/p a+2 ) · χ a+2
cyc,p ∗ ∗
∼ ⎜
⎝ 0 0 λ(β/p b+1 ) · χ b+1 ⎟
cyc,p ∗ ⎠ ,
0 0 0 λ(α)
where λ(x) is the unramified character which takes arithmetic Frobenius to x.
We remark that ρ π,p above is the contragredient of the one used in [Urb05] (we also use
the arithmetic Frobenius in defining our unramified characters), so the theorem matches
exactly with [Urb05, Cor. 1 (iii)]. Similar results in the Siegel and Klingen cases can be
found in the other parts of [Urb05, Cor. 1].
The local Galois representations appearing in Theorems 4.1.1 and 4.1.2 are sometimes
referred to as (p, p)-Galois representations. The main point of our thesis is to prove
structure theorems for the local (p, p)-Galois representations attached to automorphic
representation of GL n (A Q ).
In this study, one of the key ingredients is p-adic Hodge theory, especially, the equivalence
of categories between potentially semistable representations and filtered (ϕ, N)-
modules with coefficients and descent data.
32

4.2 p-adic Hodge theory
We start by recalling some definitions from p-adic Hodge theory. For more details, see
[Fon94], [FO], and [GM09].
Let p be a prime. Let F be a finite Galois extension of Q p and let F 0 be the maximal
unramified extension of Q p contained in F . Let E be another finite extension of Q p .
Definition 4.2.1. A filtered (ϕ, N, F, E)-module (with descent data, i.e., Gal(F/Q p )-
action) is a free of finite rank (F 0 ⊗ Qp E)-module D endowed with
1. the Frobenius endomorphism: an F 0 -semi-linear, E-linear, bijective map ϕ : D → D,
2. the monodromy operator: an F 0 ⊗ Qp E-linear, nilpotent endomorphism N : D → D
which satisfies Nϕ = pϕN,
3. an F 0 -semi-linear, E-linear action of Gal(F/Q p ) (the action on F 0 is via the projection
to Gal(F 0 /Q p )), which commutes with the action of ϕ and N,
4. a decreasing filtration (Fil i D F ) i∈Z of F ⊗ Qp E-submodules of D F = F ⊗ F0 D satisfying
Fil i D F = 0 for i ≫ 0 and Fil i D F = D F for i ≪ 0 and which are stable under
the action of Gal(F/Q p ).
For simplicity, we call a filtered (ϕ, N, F, E)-module as a filtered module or a crystal,
if the parameters are clear from the context.
4.2.1 Fontaine’s rings
We wish to recall briefly the definition of some of Fontaine’s rings of periods.
Fontaine’s rings are topological Q p -algebras B equipped with an action of G p such
that B G F
is a field for every finite extension F of Q p . For us the main example will be
B st which sits (non-canonically) between B cris and B dR . Let
Ẽ + = lim ←−
x→x p O Cp = {x = (x (0) , x (1) , ..., ) | x (i) ∈ O Cp , (x (i+1) ) p = x (i) }.
Addition given by (x + y) (i) = lim
n→∞ (x(i+n) + y (i+n) ) pn and componentwise multiplication
turn Ẽ+ into a perfect ring of characteristic p. Let Ã+ = W (Ẽ+ ) be the ring of Witt
vectors, and let ˜B + = Ã+ [1/p]. Elements of Ã+ , respectively ˜B + , may be written as
∑ p k [x k ] where the sum is over k ≥ 0, respectively k ≥ −n, for some n ≥ 0, and [ ]
denotes Teichmüller representative.
There is a ring homomorphism θ : ˜B + → C p which maps ∑ p k [x k ] to ∑ p k x (0)
k
, and
ker(θ) = (ω) is a principal ideal. We let B + dR be the completion of ˜B + with respect to
ker(θ). So elements of B + dR can be written as ∑ b n ω n with b n ∈ ˜B + .
Fix an element ɛ = (ɛ (0) , ɛ(1), . . .) ∈ Ẽ+ where ɛ (n) is a primitive p n -th root of unity.
Then θ(1 − [ɛ]) = 0 and t := log[ ] converges as a series in B + dR . Define B dR = B + dR [1/t].
Then B dR has a natural filtration given by Fil i B dR = t i B + dR
. Also, under the natural
Galois action, B G F
dR = F for any finite extension F of Q p.
Now set
B + cris = {x = ∑ w n
b n ∈ B +
n!
dR | b n ∈ ˜B + , b n → 0}
33

where the convergence of the b n is in the p-adic topology. Finally set B cris = B + cris [1/t].
Then B cris has a Frobenius ϕ : B cris → B cris , and B G F
cris = F 0.
Now let B st = B cris [Y ] be the polynomial ring in one variable over B cris . Extend
the Frobenius to B st by defining ϕ(Y ) = pY . There is also the monodromy operator
N = − d
dY
on B st which satisfies Nϕ = pϕN. Finally the Galois action on Y may be
specified, and again one has B G F
st = F 0 . Set ˜p = (p (0) , p (1) , ...) ∈ Ẽ+ where p (n) is a
primitive p n -th root of p. We will think of B st as a sub-ring of B dR by mapping Y to
log[˜p], though the definition of this last element depends on a choice, that of log p (p),
which we will take to be 0.
4.2.2 Newton and Hodge numbers
We start by recalling the definitions of Newton and Hodge numbers, and some results.
Let D be a filtered (ϕ, N, F, E)-module. Then by forgetting the E-module structure,
D is also a filtered (ϕ, N, F, Q p )-module. Let d = dim F0 D. Then Λ d F 0
D is a filtered
(ϕ, N, F, Q p )-module of dimension 1 over F 0 . Set
t H (D) = max{i ∈ Z | Fil i (F ⊗ F0 Λ d F 0
D) ≠ 0}, t N (D) = v p (λ),
where for a non-zero element x of Λ d F 0
D, ϕ(x) = λx, with λ ∈ F 0 , where v p is a normalized
valuation such that v p (p) = 1. One says that D is admissible (originally weakly
admissible), if t H (D) = t N (D), and
• for any F 0 -submodule D ′ of D stable by ϕ and N, t H (D) ≤ t N (D), where D ′ F ⊂ D F
is equipped with the induced filtration.
It turns out that the filtered (ϕ, N, F, E)-module D is admissible, if the second condition
above is replaced by the following weaker condition:
• for any (F 0 ⊗ Qp E)-submodule D ′ of D stable by ϕ and N, t H (D) ≤ t N (D), where
again D ′ F ⊂ D F is equipped with the induced filtration.
From now, we shall assume that all the conjugates of F are contained in E.
Lemma 4.2.2. Suppose D 2 ⊆ D 1 are two free of finite rank modules over F ⊗ Qp E. Then
D 1 /D 2 is also free of finite rank = rank(D 1 ) − rank(D 2 ).
Proof. We start with a basis of D 2 . For any F ⊗ Qp E-module D, we have that D ≃
∏
σ:F ↩→E D σ, where D σ = D ⊗ F ⊗E,σ E, a E-vector space. In each projection D 1σ of D 1 ,
we can extend the basis of D 2σ to D 1σ . Now pull back the extended basis vectors in each
D 1σ , we get a basis of D 1 , which extends the basis of D 2 .
Lemma 4.2.3 (Newton number). Suppose D is a filtered (ϕ, N, F, E)-module of rank n,
such that the action of ϕ is E-semisimple, i.e., there exists a basis {e 1 , · · · , e n } of D such
that ϕ(e i ) = α i e i , for some α i ∈ E × . Then
t N (D) = [E : Q p ] ·
n∑
v p (α i ).
Proof. The proof is standard from the definition of the Newton number.
34
i=1

Lemma 4.2.4 (Hodge number). Suppose D is a filtered (ϕ, N, F, E)-module of rank n.
Then
t H (D) = [E : Q p ] · ∑
i · rank F ⊗Qp E gr i D F .
i∈Z
Proof. Since D is a filtered module, there exists a filtration (Fil i D F ) i∈Z of (F ⊗ Qp E)-
submodules of D F . Then, by forgetting the E-module structure, D is also a filtered
(ϕ, N, F, Q p )-module and each term of the filtration {Fil i D F } can also thought of as an
F -module. By a standard formula (see, e.g., [FO, Prop. 6.45]), we have:
t H (D) = ∑ i∈Z
i · dim F gr i D F ,
where gr i D F = Fil i D F /Fil i+1 D F with gr i D F thought of as an F -module. By [Sav05] and
by Lemma 4.2.2, we see that each Fil i D F and gr i D F are free of finite rank over F ⊗ Qp E,
respectively. Since dim F D = [E : Q p ] · rank F ⊗Qp ED, we obtain the lemma.
Remark 4.2.5. By the last two lemmas, one can drop the common factor of [E : Q p ]
when checking the admissibility of a filtered (ϕ, N, F, E)-module.
Corollary 4.2.6. Suppose D is a filtered (ϕ, N, F, E)-module of rank 1. Then
t H (D) = [E : Q p ] · β,
where β is the unique integer such that 0 = Fil β+1 D F Fil β D F = D F .
Lemma 4.2.7. Let D 1 , D 2 be two filtered (ϕ, N, F, E)-modules, of rank r 1 , r 2 , respectively.
Assume that the action of ϕ on D 1 , D 2 is semisimple. Then
t N (D 1 ⊗ D 2 ) = rank(D 1 ) t N (D 2 ) + rank(D 2 ) t N (D 1 ),
t H (D 1 ⊗ D 2 ) = rank(D 1 ) t H (D 2 ) + rank(D 2 ) t H (D 1 ).
(4.2.1a)
(4.2.1b)
Proof. These formulas are well-known if E = Q p . The proof of (4.2.1a) is an easy check.
It is enough to prove (4.2.1b) when D 1 , D 2 are of rank 1, since
∧ r 1r 2
(D 1 ⊗ D 2 ) = ∧ r 2
(D 2 ) ⊗ · · · ⊗ ∧ r 2
(D 2 ) ⊗ ∧ r 1
(D
} {{ }
1 ) ⊗ · · · ⊗ ∧ r 1
(D 1 ),
} {{ }
r 1 -times
r 2 -times
where the tensor products are taken over F ⊗ Qp E. In this case (4.2.1b) follows from
Corollary 4.2.6.
4.2.3 Potentially semistable representations
Let V be a finite-dimensional vector space over E.
Definition 4.2.8. A representation ρ : G p → GL(V ) is said to be semistable over F , or
F -semistable, if dim F0 D st,F (V ) = dim F0 (B st ⊗ Qp V ) G F
= dim Qp V , where F 0 = B G F
st . If
such an F exists, ρ is said to be a potentially semistable representation. If F = Q p , we
say that ρ is semistable.
Remark 4.2.9. If ρ is F -semistable, then ρ is F ′ -semistable for any finite extension of
F ′ /F . Hence we may and do assume that F is Galois over Q p .
35

The following fundamental theorem plays a key role in subsequent arguments.
Theorem 4.2.10 ([CF00]). There is an equivalence of categories between F -semistable
representations ρ : G p → GL n (E) with Hodge-Tate weights −β n ≤ · · · ≤ −β 1 and admissible
filtered (ϕ, N, F, E)-module D of rank n over F 0 ⊗ Qp E such that the jumps in the
Hodge filtration Fil i D F on D F := F ⊗ F0 D are at β 1 ≤ · · · ≤ β n .
The jumps in the filtration on D F = F ⊗ F0 D st,F (ρ) are the negatives of the Hodge-
Tate weights of ρ, i.e., if h is a Hodge-Tate weight, then Fil −h+1 (D F ) Fil −h (D F ). The
equivalence of categories in the theorem is induced by Fontaine’s functor D st,F . The
Frobenius ϕ, monodromy N, and filtration on B st induce the corresponding structures on
D st,F (V ). There is also an induced action of Gal(F/Q p ) on D st,F (V ). As an illustration
of the power of the theorem we recall the following useful (and well-known) fact:
Corollary 4.2.11. Every potentially semistable character χ : G p → E × is of the form
χ = χ 0 · λ(a 0 ) · χ i cyc,p, where χ 0 is a finite order character of Gal(F/Q p ), for a cyclotomic
extension F of Q p , −i ∈ Z is the Newton number of D st,F (χ), and λ(a 0 ) is the unramified
character that takes arithmetic Frobenius to the unit a 0 = p −i /a ∈ O × E
, where a = ϕ(v)/v
for any vector v in D st,F (χ).
Proof. Every potentially semistable χ : G Q → E × is F -semistable for a sufficiently large
cyclotomic extension of Q p . Let D st,F (χ) be the corresponding filtered (ϕ, N)-module with
coefficients and descent data. Suppose that the induced Gal(F/Q p )-action on D st,F (χ)
is given the character χ 0 . Now consider the F -semistable E-valued character χ ′ := χ 0 ·
λ(a 0 ) · χ i cyc,p. One easily checks that D st,F (χ ′ ) = D st,F (χ). By Theorem 4.2.10, we have
χ = χ ′ = χ 0 · λ(a 0 ) · χ i cyc,p.
4.2.4 Weil-Deligne representations
We now recall the definition of the Weil-Deligne representation associated to an F -
semistable representation ρ : G p → GL n (E), due to Fontaine. Let W F denote the Weil
group of F , and when F = Q p , we denote W Qp by W p . For any (ϕ, N, F, E)-module D,
we have the decomposition
D ≃
[F 0 :Q p]
∏
i=1
D i , (4.2.2)
where D i = D ⊗ (F0 ⊗ Qp E,σ i ) E, and σ is the arithmetic Frobenius of F 0 /Q p .
Definition 4.2.12 (Weil-Deligne representation). Let ρ : G p → GL n (E) be an F -
semistable representation. Let D be the corresponding filtered module. Noting W p /W F =
Gal(F/Q p ), we let
g ∈ W p act on D by (g mod W F ) ◦ ϕ −α(g) , (4.2.3)
where the image of g in Gal(¯F p /F p ) is the α(g)-th power of the arithmetic Frobenius at p.
We also have the an action of N via the monodromy operator on D. These actions induce
a Weil-Deligne action on each D i in (4.2.2) and the resulting Weil-Deligne representations
are all isomorphic. This isomorphism class is defined to be the Weil-Deligne representation
WD(ρ) associated to ρ.
36

Remark 4.2.13. If F/Q p is totally ramified and Frob p ∈ W p is the arithmetic Frobenius,
then WD(ρ)(Frob p ) acts by ϕ −1 .
Lemma 4.2.14. Let ρ : G p → GL n (E) be a potentially semistable representation.
WD(ρ) is irreducible, then so is ρ.
If
Proof. Suppose the space V which affords ρ is reducible. By Theorem 4.2.10, the reducibility
of V is equivalent to the existence of a non-trivial admissible filtered (ϕ, N, F, E)-
submodule of D st,F (V ). From the definition of the Weil-Deligne action given above, this
submodule is both W p and N stable. Thus WD(ρ) is reducible, a contradiction.
4.3 The case of GL 2
Let f be a primitive ℘-ordinary form. Let ρ f,℘ be the associated Galois representation.
In this section, we shall reprove Theorem 4.1.1 (reducibility of ρ f,℘ | Gp ).
Wiles’ original proof in [Wil88] involves some amount of p-adic Hodge theory. More
precisely it uses some Dieudonné theory for the abelian varieties associated by Shimura to
cusp forms of weight 2. The result for forms of weight greater than 2 is then deduced by
a clever use of certain auxiliary Λ-adic Galois representations attached to Hida families
of ordinary forms [Hid86a] (see [BGK10, §6] for a detailed exposition of the argument).
The proof we give below avoids Hida theory completely, and (as the expert will note) is
a simple extension of Wiles’ weight 2 argument. We remark that this proof could not
have been given in [Wil88], since the equivalence of categories of Colmez and Fontaine
(Theorem 4.2.10) was of course unavailable at the time.
A key ingredient in our proof is the fact that Galois representations attached to primitive
forms live in strictly compatible systems of Galois representations [Sai97]. The consequent
ability to transfer information about the Weil-Deligne parameter between various
members of the family has been used to great effect in recent times (e.g., in the Khare-
Wintenberger proof of Serre’s conjecture) and is important for us as well. We start by
recalling the definition of such a system of Galois representations following [KW09, §5].
Let K be a number field, l be a prime, and let ρ : G K → GL n ( ¯Q l ) be a continuous
Galois representation.
Definition 4.3.1. We say that ρ is geometric, if it is unramified outside a finite set
of primes of K and its restrictions to the decomposition group at primes above l are
potentially semistable.
For every prime q of K, a geometric representation defines a representation of W ′ q,
the Weil-Deligne group of Q q , with values in GL n ( ¯Q l ), well-defined up to conjugacy.
For primes q of characteristic not l, the definition comes from the theory of Deligne-
Grothendieck, and for primes q of characteristic l, the definition comes from Fontaine
theory (cf. Definition 4.2.12).
Definition 4.3.2. For a number field L, we call an L-rational, n-dimensional strictly
compatible system of geometric representations (ρ l ) of G K the data of:
1. For each prime l and each embedding i : L ↩→ ¯Q l , a continuous, semisimple representation
ρ l : G K → GL n ( ¯Q l ) that is geometric.
37

2. For each prime q of K, an F -semisimple (Frobenius semisimple) representation
r q : W ′ q → GL n (L) such that:
• r q is unramified for all q outside a finite set.
• for each l and each i : L ↩→ ¯Q l , the Frobenius semisimple Weil-Deligne representation
W ′ q → GL n ( ¯Q l ) associated to ρ l | Gq is conjugate to r q (via the
embedding i : L ↩→ ¯Q l ).
• There are n-distinct integers β 1 < · · · < β n , such that ρ l has Hodge-Tate
weights {−β 1 , . . . , −β n } (the minus signs arise since the weights are the negatives
of the jumps in the Hodge filtration on the associated filtered module).
By work of Faltings, it is known that ρ f,℘ | Gp is a potentially semistable representation
with Hodge-Tate weights {0, k − 1}. Let D be the admissible filtered (ϕ, N, F, E)-module
associated to this representation for a suitable choices of F , and E = K f,℘ . By Theorem
4.2.10, the study of the structure of the (p, p)-Galois representation ρ f,℘ | Gp reduces
to that of the study of the filtered module D. In particular, ρ f,℘ | Gp is reducible if and
only if D has a non-trivial admissible submodule.
As mentioned above, a key ingredient in our proof of Theorem 4.1.1 is that the representation
ρ f,℘ lives in a strictly compatible system of Galois representations (ρ f,λ ), where
λ varies over the primes of K f . This result is the culmination of the work of several
people, including Langlands, Deligne, Carayol, Katz-Messing, and most recently Saito
[Sai97]. In particular, one may read off the Weil-Deligne representation WD(ρ f,℘ | Gp ) on
D (cf. Definition 4.2.12) by looking at the Weil-Deligne representation attached to ρ f,λ | Gp
for a place λ of K f with λ ∤ p. As a consequence, one may read off, e.g., the characteristic
polynomial of crystalline Frobenius ϕ purely in terms of a λ-adic member of the strictly
compatible family for λ ∤ p.
Let π be the cuspidal automorphic representation of GL 2 (A Q ) corresponding to f. It is
well-known that f is ordinary at ℘ only if the underlying local automorphic representation
π p is in the principal series or (an unramified twist of) the Steinberg representation. We
obtain:
Theorem 4.3.3. The characteristic polynomial P (X) of the (inverse of) crystalline
Frobenius ϕ on D coincides with that of ρ f,λ (Frob p ) for a place λ of K f with λ ∤ p.
More precisely, in the cases we need, P (X) is given by:
• (Unramified principal series) If p ∤ N, then ρ f,℘ | Gp is crystalline and P (X) =
X 2 − a p (f)X + χ(p)p k−1 .
• (Steinberg) If p||N and v p (cond(χ)) = 0, then ρ f,℘ | Gp is Q p -semistable and P (X) =
(X − a p (f)) (X − pa p (f)).
• (Ramified principal series) If p|N with v p (N) = v p (cond(χ)), then ρ f,℘ | Gp is potentially
crystalline and P (X) = (X − a p (f))(X − χ ′ (p)ā p (f)), where χ ′ denotes the
prime-to-p part of the character χ.
More generally, the complete rank 2 filtered (ϕ, N, F, E)-module D can be written down
quite explicitly in all the above cases (cf. [Bre01], [GM09]). We now give a proof of
Theorem 4.1.1 using this structure of D, which we first recall below in various cases.
38

Good reduction: p ∤ N
In this case F = Q p and ρ f,℘ | Gp is crystalline (N = 0). Let D be the corresponding filtered
(ϕ, N)-module. Let α and β be the two roots of P (X) = X 2 − a p (f)X + χ(p)p k−1 , with
v p (α) = 0 and v p (β) = k − 1, since v p (a p (f)) = 0. The structure of the filtered module
D is well-known (cf. [Bre01, p. 30-32], where the normalizations are a bit different).
There are essentially two possibilities for D depending on whether D is decomposable or
indecomposable. If D is decomposable, then D = Ee 1 ⊕ Ee 2 , and
D =
{
ϕ(e1 ) = α −1 e 1 , ϕ(e 2 ) = β −1 e 2 ,
Fil i (D F ) = 0 if i ≥ 1, Ee 1 if 2 − k ≤ i ≤ 0, D if i ≤ 1 − k.
If D is indecomposable, then D = Ee 1 ⊕ Ee 2 , and
D =
{
ϕ(e1 ) = α −1 e 1 + p 1−k e 2 , ϕ(e 2 ) = β −1 e 2 ,
Fil i (D F ) = 0 if i ≥ 1, Ee 1 if 2 − k ≤ i ≤ 0, D if i ≤ 1 − k.
Steinberg case: p ‖ N and p ∤ cond(χ)
In this case F = Q p , ρ f,℘ | Gp is semistable over Q p but non-crystalline, and v p (a p (f)) =
k−2
2 . Note v p(a p (f)) = 0 if and only if k = 2. Let D be the corresponding filtered
(ϕ, N)-module (cf. [GM09, §3.1]). Set α = a p (f) and β = pa p (f). Then D = Ee 1 ⊕ Ee 2 ,
and
⎧
⎪⎨
ϕ(e 1 ) = pβ −1 e 1 , ϕ(e 2 ) = β −1 e 2 ,
D = N(e 1 ) = e 2 , N(e 2 ) = 0,
⎪⎩
Fil i (D F ) = 0 if i ≥ 1, E(e 1 − Le 2 ) if 2 − k ≤ i ≤ 0, D if i ≤ 1 − k,
for some unique non-zero L ∈ E.
Ramified principal series: m = v p (cond(χ)) = v p (N) ≥ 1
In this case, ρ f,℘ | Gp becomes crystalline over the totally ramified abelian extension F =
Q p (µ p m) of Q p . Decompose χ = χ p χ ′ into its p-part and prime-to-p part. Let D be the
associated admissible filtered (ϕ, N, F, E)-module. An explicit description of this module
was given in [GM09, §3.2]. Set α = a p (f) and β = χ ′ (p)ā p (f). Then D = Ee 1 ⊕ Ee 2 , and
D =
{
ϕ(e1 ) = α −1 e 1 , ϕ(e 2 ) = β −1 e 2 ,
g(e 1 ) = e 1 , g(e 2 ) = χ p (g)e 2 ,
for g ∈ Gal(F/Q p ). Since v p (a p (f)) = 0, the module D is either D ord-split or D ord-non-split
(cf. [GM09, §3.2]). The corresponding filtrations in these cases are given by
Fil i (D F )
= 0 if i ≥ 1, (F ⊗ E)e 1 if 2 − k ≤ i ≤ 0, D F if i ≤ 1 − k, and,
Fil i (D F ) = 0 if i ≥ 1, (F ⊗ E)(xe 1 + ye 2 ) if 2 − k ≤ i ≤ 0, D F if i ≤ 1 − k,
respectively, where x and y are explicit non-zero quantities in F ⊗ E (cf. [GM09, §3.2]).
39

4.3.1 Proof of Wiles’ theorem
Let D n be the submodule of D generated by e n , for n = 1, 2. Since, in all cases, the ‘line’
which determines the interesting step in the filtration on D F , is transverse to D 2,F , the
induced filtration on D 2,F is given by
· · · = Fil 2−k (D 2,F ) = 0 Fil 1−k (D 2,F ) = D 2,F (4.3.1)
Thus t H (D 2 ) = 1 − k in all cases.
In the principal series cases (i.e., the first and third cases above), we see that t N (D 2 ) =
v p (β −1 ) = 1−k, so that D 2 is an admissible ϕ-submodule of D. In the Steinberg case, D 2 is
the only (ϕ, N)-submodule of rank 1, since N(e 2 ) = 0. Moreover, t N (D 2 ) = v p (β −1 ) = − k 2
which equals 1 − k = t H (D 2 ) if and only if k = 2. Thus D 2 is an admissible submodule
if and only if k = 2, and D is irreducible if k > 2. Thus, in all (ordinary) cases, we have
shown the existence of an admissible submodule D 2 of D, associated to ρ f,℘ | Gp , such that
on D/D 2 , crystalline Frobenius ϕ acts by an explicit element α −1 of valuation zero.
Assume that we are in the first two cases, so that F = Q p . By Theorem 4.2.10, the
representation ρ f,℘ | Gp is clearly reducible, with a one-dimensional submodule given by the
character λ(β/p k−1 )χ k−1
cyc,p, and quotient given by the unramified character λ(α) (see Corollary
4.2.11), proving the theorem in these cases. In the last case (when F ≠ Q p ), again
by Theorem 4.2.10, ρ f,℘ | Gp is reducible. Indeed, the module D 2 is a filtered (ϕ, N, F, E)-
module with descent data given by the character χ p of Gal(F/Q p ). By Corollary 4.2.11,
D 2 corresponds to the character ψ = χ p λ(β/p k−1 )χ k−1
cyc,p of G p . Since D 2 ⊂ D as filtered
modules with descent data, we see V (ψ) is a one dimensional submodule of ρ f,℘ | Gp with
unramified quotient given by λ(α), proving the theorem in this case as well.
4.4 The case of GL n
In the next chapter, we prove various generalizations of Theorem 4.1.1 for the local (p, p)-
Galois representations attached to automorphic representation of GL n (A Q ).
In this section we collect together some facts about such automorphic representations
and their Galois representations needed for the proof. The main results we need are the
Local Langlands correspondence (now a theorem of Henniart [Hen00] and Harris-Taylor
[HT01]), and the existence of a strictly compatible systems of Galois representations
attached to cuspidal automorphic representation of GL n (much progress has been made
on this by Clozel, Harris, Kottwitz and Taylor [CHT08]).
4.4.1 Local Langlands correspondence
We state a few results concerning the Local Langlands correspondence. We follow Kudla’s
article [Kud94], noting this article follows Rodier [Rod82], which in turn is based on the
original work of Bernstein and Zelevinsky.
Let F be a complete non-Archimedean local field of residue characteristic p, let n ≥ 1,
and let G = GL n (F ). For a partition n = n 1 +n 2 +· · ·+n r of n, let P be the corresponding
parabolic subgroup of G, M the Levi subgroup of P , and N the unipotent radical of P . Let
δ P denote the modulus character of the adjoint action of M on N. If σ = σ 1 ⊗σ 2 ⊗· · ·⊗σ r
40

is a smooth representation of M on V , we let
I G P (σ) = {f : G → V | f smooth on G and f(nmg) = δ 1 2
P
(m)(σ(m)(f(g))},
for n ∈ N, m ∈ M, g ∈ G. G acts on functions in IP G(σ) by right translation and IG P
(σ) is
the usual induced representation of σ. It is an admissible representation of finite length.
A result of Bernstein-Zelevinsky says that if all the σ i supercuspidal, and σ is irreducible,
smooth and admissible, then IP G(σ) is reducible if and only if n i = n j and
σ i = σ j (1) for some i ≠ j. For the partition n = m + m + · · · + m (r times), and for a
supercuspidal representation of σ of GL m (F ), call the data
(σ, σ(1), . . . , σ(r − 1)) = [σ, σ(r − 1)] = ∆
a segment. Clearly IP G (∆) is reducible. By [Kud94, Thm. 1.2.2], the induced representation
IP G (∆) has a unique irreducible quotient Q(∆), which is essentially square-integrable.
Two segments
∆ 1 = [σ 1 , σ 1 (r 1 − 1)] and ∆ 2 = [σ 2 , σ 2 (r 2 − 1)]
are said to be linked, if ∆ 1 ∆ 2 , ∆ 2 ∆ 1 , and ∆ 1 ∪ ∆ 2 is a segment. We say that ∆ 1
precedes ∆ 2 if ∆ 1 and ∆ 2 are linked and if σ 2 = σ 1 (k), for some positive integer k.
Theorem 4.4.1 (Langlands classification). Given segments ∆ 1 , . . . , ∆ r , assume that for
i < j, ∆ i does not precede ∆ j . Then
1. The induced representation IP G(Q(∆ 1) ⊗ · · · ⊗ Q(∆ r )) admits a unique irreducible
quotient Q(∆ 1 , . . . , ∆ r ), called the Langlands quotient. Moreover, r and the segments
∆ i up to permutation are uniquely determined by the Langlands quotient.
2. Every irreducible admissible representation of GL n (F ) is isomorphic to the Langlands
quotient Q(∆ 1 , . . . , ∆ r ), for some ∆ i ’s.
3. The induced representation I G P (Q(∆ 1) ⊗ · · · ⊗ Q(∆ r )) is irreducible if and only if no
two of the segments ∆ i and ∆ j are linked.
We now turn to the Galois side.
Representations of the Weil-Deligne group
A representation of W F is said to be Frobenius semisimple, if the arithmetic Frobenius
acts semisimply. An admissible representation of W
F ′ , the Weil-Deligne group of F , is one
for which the action of W F is Frobenius semisimple. Let Sp(r) denote the Weil-Deligne
representation of order r with the usual definition. When F = Q p , there is a basis {f i }
of Sp(r) for which ϕf i = p i−1 f i , and Nf i = f i−1 for i > 1 and Nf 1 = 0.
Deligne classified all such admissible representations of W
F ′ as the direct sum of the
indecomposable representations τ m ⊗ Sp(r), where τ m is an irreducible admissible representation
of dimension m of W F and r ≥ 1 (cf. [Roh94, §5, Cor. 2]).
Theorem 4.4.2 (Local Langlands correspondence). ([HT01, VII.2.20], [Hen00], [Kut80]).
There exists a bijection between isomorphism classes of irreducible admissible representations
of GL n (F ) and isomorphism classes of admissible n-dimensional representations of
W ′ F , the Weil-Deligne group of F . 41

The correspondence is given as follows. The key point is to construct a bijection Φ F
between the set of isomorphism classes of supercuspidal representations of GL n (F ) and
the set of isomorphism classes of irreducible admissible representations of W F . This is
due to Henniart [Hen00] and Harris-Taylor [HT01]. Then, to Q(∆), for the segment ∆ =
[σ, σ(r−1)], one associates the indecomposable admissible representation Φ F (σ)⊗Sp(r) of
W
F ′ . More generally, to the Langlands quotient Q(∆ 1, . . . , ∆ r ), where ∆ i = [σ i , σ i (r i −1)],
for i = 1, . . . , k, one associates the admissible representation ⊕ k i=1 Φ F (σ i ) ⊗ Sp(r i ) of W
F ′ .
4.4.2 Automorphic forms on GL n
We now briefly recall what are (cuspidal) automorphic forms on GL n (A Q ). More details
can be found in [Tay04].
The Harish-Chandra isomorphism identifies the center z n of the universal enveloping
algebra of the complexified Lie algebra gl n of GL n , with the commutative C-algebra
C[X 1 , X 2 , . . . , X n ] Sn , where the symmetric group S n acts by permuting the X i . Given a
multiset H = {x 1 , x 2 , . . . , x n } of n complex numbers one obtains an infinitesimal character
of z n give by χ H : X i ↦→ x i .
Cuspidal automorphic forms with infinitesimal character χ H (or more simply just H)
are smooth functions f : GL n (Q)\GL n (A Q ) → C satisfying the usual finiteness condition
under a maximal compact subgroup, a cuspidality condition, and a growth condition for
which we refer the reader to [Tay04]. In addition, if z ∈ z n then z ·f = χ H (z)f. The space
of such functions is denoted by A ◦ H (GL n(Q)\GL n (A Q )). This space is a direct sum of irreducible
admissible GL n (A (∞)
Q
)×(gl n, O(n))-modules each occurring with multiplicity one,
and these irreducible constituents are referred to as cuspidal automorphic representation
of GL n (A Q ) with infinitesimal character χ H . Let π be such an automorphic representation.
By a result of Flath, π is a restricted tensor product π = ⊗ ′ pπ p (cf. [Bum97, Thm.
3.3.3]) of local automorphic representations.
4.4.3 Galois representations
Let π be an automorphic representation of GL n (A Q ) with infinitesimal character χ H ,
where H is a multiset of integers. The following very strong, but natural, conjecture
seems to be part of the folklore.
Conjecture 4.4.3. Let H consist of n distinct integers. There is a strictly compatible
system of Galois representations (ρ π,l ) associated to π, with Hodge-Tate weights H, such
that Local-Global compatibility holds.
Here Local-Global compatibility means that the underlying semisimplified Weil -
Deligne representation at p in the compatible system (which is independent of the residue
characteristic l of the coefficients by hypothesis) corresponds to π p via the Local Langlands
correspondence. Considerable evidence towards this conjecture is available for self-dual
representations thanks to the work of Clozel, Kottwitz, Harris and Taylor. We quote the
following theorem from Taylor’s paper [Tay04], referring to that paper for the original
references (e.g., [Clo91]).
Theorem 4.4.4 (cf. [Tay04], Thm. 3.6). Let H consist of n distinct integers. Suppose
that the contragredient representation π ∨ = π ⊗ ψ for some character ψ : Q × \A × Q → C× ,
42

and suppose that for some prime q, the representation π q is square-integrable. Then there
is a continuous representation
ρ π,l : G Q → GL n ( ¯Q l )
such that ρ π,l | Gl is potentially semistable with Hodge-Tate weights given by H, and such
that for any prime p ≠ l, the semisimplification of the Weil-Deligne representation attached
to ρ π,l | Gp is the same as the Weil-Deligne representation associated by the Local
Langlands correspondence to π p , except possibly for the monodromy operator.
Taylor and Yoshida [TY07] show that the two Weil-Deligne representations in the
theorem above are in fact the same (i.e., the monodromy operators also match).
In any case, from now on, we shall assume that Conjecture 4.4.3 holds. In particular,
we assume that the Weil-Deligne representation at p associated to a p-adic member of the
compatible system of Galois representations attached to π using Fontaine theory is the
same as the Weil-Deligne representation at p attached to an l-adic member of the family,
for l ≠ p.
4.4.4 A variant, following [CHT08]
A variant of the above result can be found in [CHT08]. We state this now using the
notation and terminology from [CHT08, §4.3].
Say π is an RAESDC (regular, algebraic, essentially self dual, cuspidal) automorphic
representation if π is a cuspidal automorphic representation such that
• π ∨ = π ⊗ χ for some character χ : Q × \A × Q → C× .
• π ∞ has the same infinitesimal character as some irreducible algebraic representation
of GL n .
Let a ∈ Z n satisfy
a 1 ≥ · · · ≥ a n . (4.4.1)
Let Ξ a denote the irreducible algebraic representation of GL n with highest weight a. We
say that an RAESDC automorphic representation π has weight a, if π ∞ has the same
infinitesimal character as Ξ ∨ a . In this case there is an integer w a such that a i +a n+1−i = w a
for all i.
Let S be a finite set of primes of Q. For v ∈ S, let ρ v be an irreducible square
integrable representation of GL n (Q v ). Say that an RAESDC representation π has type
{ρ v } v∈S , if for each v ∈ S, π v is an unramified twist of ρ ∨ v .
With this setup, Clozel, Harris, and Taylor attached a Galois representation to an
RAESDC π.
Theorem 4.4.5 ([CHT08], Prop. 4.3.1). Let ι : ¯Q l ≃ C. Let π be an RAESDC automorphic
representation as above of weight a and type {ρ v } v∈S . Then there is a continuous
semisimple Galois representation
with the following properties:
r l,ι (π) : Gal( ¯Q/Q) → GL n ( ¯Q l )
43

1. For every prime p ∤ l, we have
r l,ι (π)| ss
G p
= (r l (ι −1 π p ) ∨ )(1 − n) ss ,
where r l is the reciprocity map defined in [HT01].
2. If l = p, then the restriction r l,ι (π)| Gp is potentially semistable and if π p is unramified
then it is crystalline, with Hodge-Tate weights −(a j + n − j) for j = 1, . . . , n.
4.4.5 Newton and Hodge filtration
Let ρ π,p | Gp be the (p, p)-Galois representation attached to an automorphic representation
π and D be the corresponding filtered (ϕ, N, F, E)-module (for suitable choices of F and
E).
Note that there are are two natural filtrations on D F , the Hodge filtration Fil i D F
and the Newton filtration defined by ordering the slopes of the crystalline Frobenius (the
valuations of the roots of ϕ.) To keep the analysis of the structure of the (p, p)-Galois
representation ρ π,p | Gp within reasonable limits in this thesis, we shall make the following
assumption.
Assumption 4.4.6. The Newton filtration on D F is in general position (or transverse)
with respect to the Hodge filtration Fil i D F .
Here, if V is a space and Fil i 1V and Fil j 2V are two filtrations on V then we say they
are in general position if the each Fil i 1V is as transverse as possible to each Fil j 2 V . (cf.
(4.3.1)). We remark that the condition above is in some sense generic, since two random
filtrations on a space tend to be in general position.
We end this chapter by recalling the notion of ordinary and quasi-ordinary for p-adic
representations.
4.4.6 Ordinary and quasi-ordinary representations
As mentioned earlier, our goal is to prove that the (p, p)-Galois representation attached to
π is ‘upper-triangular’ in several cases. To this end it is convenient to recall the following
terminology (see, e.g., Greenberg [Gre94, p. 152] or Ochiai [Och01, Def. 3.1]).
Definition 4.4.7. Let F be a number field. A p-adic representation V of G F is called
ordinary (respectively quasi-ordinary), if the following conditions are satisfied:
1. For each places v of F over p, there is a decreasing filtration of G Fv -modules
· · · Fil i vV ⊇ Fil i+1
v V ⊇ · · ·
such that Fil i vV = V for i ≪ 0 and Fil i vV = 0 for i ≫ 0.
2. For each v and each i, I v acts on Fil i vV/Fil i+1
v V via the character χ i cyc,p, where χ cyc,p
is the p-adic cyclotomic character (respectively, there exists an open subgroup of I v
which acts on Fil i vV/Fil i+1
v V via χ i cyc,p).
44

Chapter 5
On Local Galois Representations
attached to Automorphic Forms
This chapter contains the main results of the first part of the thesis. We prove structure
theorems for the local Galois representations attached to cuspidal automorphic representation
of GL n (A Q ).
5.1 Introduction
Recall that we have assumed that the global p-adic Galois representation ρ π,p attached
to π exists, and it satisfies several natural properties, e.g., it lives in a strictly compatible
system of Galois representations, and satisfies Local-Global compatibility. Recently, much
progress has been made on this front: such Galois representations have been attached to
what are referred to as RAESDC automorphic representation of GL n (A Q ) by Clozel,
Kottwitz, Harris and Taylor, and for conjugate self-dual automorphic representations
over CM fields these representations were shown to satisfy Local-Global compatibility by
Taylor-Yoshida, away from p.
Under some standard hypotheses (e.g., Assumption 4.4.6), we show that in several
cases the corresponding local representation ρ π,p | Gp has an ‘upper-triangular’ form, and
completely determine the ‘diagonal’ characters. In other cases, and perhaps more interestingly,
we give conditions under which this local representation is irreducible. For instance,
we directly generalize the comment about irreducibility made just after Theorem 4.1.1.
One of the main theorems is proved in §5.3 and §5.4, using methods from p-adic Hodge
theory. It is well-known that the category of Weil-Deligne representations is equivalent
to the category of (ϕ, N)-modules [BS07, Prop. 4.1]. In §5.4, we classify the (ϕ, N)-
submodules of the (ϕ, N)-module associated to the indecomposable Weil-Deligne representation
is the theorem. This classification plays a key role in analyzing the (p, p)-Galois
representation once the Hodge filtration is introduced. Along the way, we take a slight
detour to write down explicitly the filtered (ϕ, N)-module attached to an m-dimensional
‘unramified supercuspidal’ representation, since this might be a useful addition to the
literature (cf. [GM09] for the two-dimensional case).
Some results in the decomposable case (where the Weil-Deligne representation is a
direct sum of indecomposables) are given in §5.5, though the principal series case is
45

treated completely a bit earlier, in §5.2 (in the spherical case our results overlap with
those in D. Geraghty’s recent thesis).
5.2 Principal series
Let π be a cuspidal automorphic representation of GL n (A Q ) with infinitesimal character
H, for a set of distinct integers H. In this section, we study the behaviour of the (p, p)-
Galois representation assuming that π p , the local automorphic representation, is in the
principal series.
5.2.1 Spherical case
Assume that π p is an unramified principal series representation. Then there exist unramified
characters χ 1 , . . . , χ n of Q × p such that π p is the Langland’s quotient Q(χ 1 , . . . , χ n ).
We can parametrize the isomorphism class of this representation by the Satake parameters
α 1 , . . . , α n , for α i = χ i (ω), where ω is a uniformizer for Q p .
Note that ρ π,p | Gp is crystalline with Hodge-Tate weights H. Let D be the corresponding
filtered ϕ-module. Let the jumps in the filtration on D be β 1 < β 2 < · · · < β n (so
that the Hodge-Tate weights H are −β 1 > · · · > −β n .
Definition 5.2.1. Say that the automorphic representation π is p-ordinary, if β i +
v p (α i ) = 0, for all i = 1, . . . , n.
Theorem 5.2.2 (Spherical case). Let π be an automorphic representation of GL n (A Q )
with infinitesimal character given by the integers −β 1 > · · · > −β n and such that π p is
in the unramified principal series with Satake parameters α 1 , . . . , α n . If π is p-ordinary,
then ρ π,p | Gp ∼
⎛
α
λ( 1
) · χ −β ⎞
1
p vp(α 1 ) cyc,p ∗ · · · ∗
α 0 λ( 2
) · χ −β 2
p vp(α 2 ) cyc,p · · · ∗
0 0 · · · ∗
.
⎜
α
⎝
0 0 λ( n−1
) · χ −β n−1
p vp(α n−1 ) cyc,p
∗ ⎟
⎠
α
0 0 0 λ( n
)) · χ −βn
p vp(αn) cyc,p
In particular, ρ π,p | Gp
is ordinary.
Proof. Since π p is p-ordinary, we have that v p (α n ) < v p (α n−1 ) < · · · < v p (α 1 ). By strict
compatibility, the characteristic polynomial of the inverse of crystalline Frobenius of D n
is equal to ∏ i (X − α i).
Since the v p (α i ) are distinct, there exists a basis of eigenvectors of D n for the operator
ϕ, say {e i }, with corresponding eigenvalues {αi
−1 }. For any integer 1 ≤ i ≤ n, let D i be
the ϕ-submodule generated by {e 1 , . . . , e i }. The filtration on D n is
· · · ⊆ 0 Fil βn (D n ) ⊆ · · · Fil β 1
(D n ) = D n ⊆ · · · (5.2.1)
Since D n is admissible, we have that
n∑
β i = −
i=1
46
n∑
v p (α i ). (5.2.2)
i=1

Clearly, we have that the jumps in the induced filtration on D n−1 are β 1 , . . . , β n−1 . By
(5.2.2), we have
n−1
∑
n−1
∑
t H (D n−1 ) = β i = − v p (α i ) = t N (D n−1 ), (5.2.3)
i=1
since β n = −v p (α n ) and hence the module D n−1 is admissible. Moreover, D n /D n−1 is
also admissible, because t H (D n /D n−1 ) = β n = −v p (α n ) = t N (D n /D n−1 ), since ϕ acts on
D n /D n−1 by αn −1 . Therefore, the Galois representation ρ π,p | Gp looks like
( )
ρn−1 ∗
ρ ∼
α
0 λ( n
,
) · χ −βn
p vp(αn) cyc,p
where ρ n−1 is the (n − 1)-dimensional representation of G p which corresponds to D n−1 .
Repeating this argument successively for D n−1 , D n−2 , . . . D 1 , we obtain the theorem.
Corollary 5.2.3. When n = 2, and π corresponds to f, we recover part (i) of Theorem
4.1.1 (at least when k ≥ 3 is odd).
Proof. Say 0 ≤ v p (α) ≤ v p (β) ≤ k − 1. It is well-known (cf. [Bum97]) that the Satake
parameters at p satisfy the formulas α 1 = β · p 1−k
2 and α 2 = α · p k−1
2 . In fact L(s, π) =
L(s + k−1
2 , f) and ρ π,p = ρ f,℘ ⊗ χ 1−k
2
cyc,p, where (k − 1)/2 ∈ Z if k is odd. In particular, the
Hodge-Tate weights of ρ π,p | Gp are −β 2 = 1−k
2
and −β 1 = k−1
2
, distinct integers if k ≥ 3 is
odd. By the p-ordinariness condition for π, we have that v p (α 2 ) = 1−k
2
and v p (α 1 ) = k−1
2 .
By the theorem above, we obtain
⎛
⎞
ρ π,p | Gp ∼ ⎝ λ(α 1/p (k−1)/2 ) · χ k−1
2
cyc,p
∗
⎠ .
0 λ(α 2 /p (1−k)/2 ) · χ 1−k
2
cyc,p
Twisting both sides by χ k−1
2
cyc,p, we recover part (i) of Theorem 4.1.1.
Variant, following [CHT08]
Let π now be an RAESDC representation of weight a as in §4.4.4 and let π p denote
the local p-adic automorphic representation associated to π. For any i = 1, . . . , n, set
β n+1−i ′ := a i + n − i, where a i ’s are as in (4.4.1). We have that β n ′ > β n−1 ′ > · · · > β′ 1 ,
and the Hodge-Tate weights are −β n ′ < −β n−1 ′ < · · · < −β′ 1 .
Assume that π p is in the unramified principal series, so π p = Q(χ 1 , χ 2 , . . . , χ n ), where
χ i ’s are unramified characters of Q × p . Set α i ′ = χ i(ω)p n−1
2 . Let t (j)
p denote the eigenvalue
of T p
(j) on πp
GLn(Zp) , where T p
(j) is j-th Hecke operator as in [CHT08], and πp
GLn(Zp) is
spanned by GL n (Z p )-fixed vector, unique up to a constant. We would like to compute
the right hand side in the display in part (1) of Theorem 4.4.5. By [CHT08, Cor. 3.1.2],
in the spherical case, one has
(r l (ι −1 π p ) ∨ )(1 − n)(Frob −1
p ) = ∏ i
i=1
(X − α ′ i)
= X n − t (1)
p X n−1 + · · · + (−1) j p j(j−1)
2 t (j)
p
X n−j + · · · + (−1) n p n(n−1)
2 t (n)
p ,
47

where Frob −1
p is geometric Frobenius. Let s j denote the j-th elementary symmetric polynomial.
Then from the equation above, for any j = 1, · · · , n, we have p j(j−1)
2 t (j)
p =
s j (α i ′) = p j(n−1)
2 s j (χ i (p)) and hence t (j)
p = s j (χ i (p))p j(n−j)
2 . In this setting we make:
Definition 5.2.4. Say that the automorphic representation π p is p-ordinary if β i ′ +
v p (α i ′ ) = 0 for all i = 1, · · · , n.
Note that by strict compatibility, crystalline Frobenius has characteristic polynomial
exactly that above. The following theorem follows in a manner identical to that used to
prove Theorem 5.2.2.
Theorem 5.2.5 (Spherical case, variant). Let π be a cuspidal automorphic representation
of GL n (A Q ) of weight a, as in §4.4.4. Let r p,ι (π) be the corresponding p-adic Galois
representation, with Hodge-Tate weights −β n+1−i ′ := a i + n − i, for i = 1, . . . , n. Suppose
π p is in the principal series with Satake parameters α 1 , . . . , α n , and set α i ′ = α ip n−1
2 . If
π is p-ordinary, then r p,ι (π)| Gp ∼
⎛
α
λ(
′ 1
) · ⎞
p vp(α′ 1 ) χ−β′ 1
cyc,p ∗ · · · ∗
α 0 λ(
′ 2
) · p vp(α′ 2 ) χ−β′ 2
cyc,p · · · ∗
.
⎜
⎝ 0 0 · · · ∗ ⎟
⎠
α
0 0 0 λ(
′ n
)) · p vp(α′ n ) χ−β′ n
cyc,p
In particular, r p,ι (π)| Gp
is ordinary.
The result above was also obtained by D. Geraghty in the course of proving modularity
lifting theorems for GL n (see Lem. 2.7.7 and Cor. 2.7.8 of [Ger10]).
5.2.2 Ramified principal series case
Returning to the case where π is an automorphic representation with infinitesimal character
H, we assume now that π p = Q(χ 1 , . . . , χ n ), where χ i are possibly ramified characters
of Q × p .
By the Local Langlands correspondence, we can think of the χ i as characters of W p .
In particular, χ i | Ip have finite image. By strict compatibility, WD(ρ)| Ip ≃ ⊕ i χ i| Ip . The
characters χ i | Ip factor through Gal(Q nr
p (ζ p m)/Q nr
p ) ≃ Gal(Q p (ζ p m)/Q p ) for some m ≥ 1.
Set F := Q p (ζ p m). Observe that F is a finite abelian totally ramified extension of Q p .
Note that ρ π,p | GF is crystalline. Let D be the corresponding filtered module. Then
D = Ee 1 + · · · + Ee n , where g ∈ Gal(F/Q p ) acts by χ i on e i . A small computation shows
that ϕ(e i ) = αi −1 e i , where α i = χ i (ω F ), for ω F a uniformizer of F . Using Corollary 4.2.11,
and following the proof of Theorem 5.2.2, we obtain:
Theorem 5.2.6 (Ramified principal series). Say π p = Q(χ 1 , . . . , χ n ) is in the ramified
principal series. If π is p-ordinary, then ρ π,p | Gp ∼
⎛
α
χ 1 · λ( 1
) · χ −β ⎞
1
p vp(α 1 ) cyc,p ∗ · · · ∗
α
0 χ 2 · λ( 2
) · χ −β 2
p
⎜
vp(α 2 ) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ .
α
0 0 0 χ n · λ( n
)) · χ −βn
p vp(αn) cyc,p
48

In particular, ρ π,p | Gp is quasi-ordinary.
5.3 Steinberg case
In this section we treat the case where the Weil-Deligne representation attached to π p is
a twist of the special representation Sp(n).
5.3.1 Unramified twist of Steinberg
We start with the case where the Weil-Deligne representation attached to π p is of the
form χ ⊗ Sp(n), where χ is an unramified character.
Let D be the filtered (ϕ, N, Q p , E)-module attached to ρ π,p | Gp . Thus D is a vector
space over E. Note N n = 0 and N n−1 ≠ 0 so that there is a basis {f i } of D with
f i−1 := Nf i , for 1 < i ≤ n, and Nf 1 = 0. We set α := χ(Frob p ). Since Nϕ = pϕN,
we may assume that ϕ(f i ) = αi
−1 f i , ∀i, where αi
−1 = pi−1
α
. When α = 1, D = Sp(n) as
mentioned in §4.4.1.
For each 1 ≤ i ≤ n, let D i denote the subspace 〈f i , . . . , f 1 〉. Clearly, dim(D i ) = i and
D 1 D 2 · · · D n = D. We have:
Lemma 5.3.1. For every integer 1 ≤ r ≤ n, there is a unique N-submodule of D of rank
r, namely D r .
Proof. Let D ′ be a N-submodule of D of rank r. Say the order of nilpotency of N on
D ′ is i. Thus, D ′ ⊆ Ker(N i ). Since Ker(N i ) = 〈f i , . . . , f 1 〉, dim(Ker(N i )) = i and hence
r ≤ i. Clearly, the order of nilpotency of N on D ′ is less than or equal to r. Hence i = r
and D ′ = Ker(N r ) = D r .
Let β n > · · · > β 1 be the jumps in the Hodge filtration on D. We assume that the
Hodge filtration is in general position with respect to the Newton filtration given by the
D i (cf. Assumption 4.4.6). An example of such a filtration is
〈f n 〉 〈f n , f n−1 〉 · · · 〈f n , . . . , f 2 〉 〈f n , . . . , f 1 〉.
The following elementary lemma plays an important role in later proofs.
Lemma 5.3.2. Let m be a natural number. Let {a i } n i=1 be an increasing (resp., decreasing)
sequence of integers such that |a i+1 −a i | = m. Let {b i } n i=1 be another increasing (resp.,
decreasing) sequence of integers, such that |b i+1 − b i | ≥ m. Assume that ∑ i a i = ∑ i b i.
If a n = b n or a 1 = b 1 , then a i = b i , ∀ i.
Proof. Let us prove the lemma when a n = b n and a i are increasing. The proof in the
other cases is similar. We have:
m(n − 1 + n − 2 + · · · + 1) ≤
n∑
(b n − b i ) =
i=1
n∑
(a n − a i ) = m(n − 1 + n − 2 + · · · + 1)
The first equality in the above expression follows from a n = b n . From the above equation,
we see that b n − b i = a n − a i , for every 1 ≤ i ≤ n. Since a n = b n , we have that a i = b i ,
for every 1 ≤ i ≤ n.
i=1
49

By Lemma 5.3.1, the D i are the only (ϕ, N)-submodules of D. The following proposition
shows that if two ‘consecutive’ submodules D i and D i+1 are admissible then all the
D i are admissible.
Proposition 5.3.3. Suppose there exists an integer 1 ≤ i ≤ n such that both D i and
D i+1 are admissible. Then each D r , for 1 ≤ r ≤ n, is admissible. Moreover, the β i are
consecutive integers.
Proof. Since D i and D i+1 are admissible, we have the following equalities:
i∑
β 1 + β 2 + · · · + β i = − v p (α r ),
r=1
i+1
∑
β 1 + β 2 + · · · + β i+1 = − v p (α r ).
r=1
(5.3.1)
Define a r = −v p (α r ) and b r = β r , for 1 ≤ r ≤ n. Hence, we have
a n > · · · > a i+2 > a i+1 > a i > · · · > a 1 ,
b n > · · · > b i+2 > b i+1 > b i > · · · > b 1 .
(5.3.2)
By (5.3.1), we have a i+1 = −v p (α i+1 ) = β i+1 = b i+1 . By Lemma 5.3.2 and by (5.3.1), we
have that a r = b r , for all 1 ≤ r ≤ i + 1. Since D n is admissible, we have
t H (D n ) =
n∑
β r = −
r=1
From (5.3.1) and (5.3.3), we have that
n∑
r=i+1
β r = −
n∑
v p (α r ) = t N (D n ). (5.3.3)
r=1
n∑
r=i+1
v p (α r ).
Again, by (5.3.2) and Lemma 5.3.2, we have that a r = b r , for all i + 1 ≤ r ≤ n. Hence
β r = −v p (α r ), for all 1 ≤ r ≤ n. This shows that all other D i ’s are admissible. Also, the
β i are consecutive integers since the v p (α i ) are consecutive integers.
Corollary 5.3.4. Keeping the notation as above, the admissibility of D 1 or D n−1 implies
the admissibility of all other D i .
Theorem 5.3.5. Assume that the Hodge filtration on D is in general position with respect
to the D i (cf. Assumption 4.4.6). Then the crystal D is either irreducible or reducible,
in which case each D i , for 1 ≤ i ≤ n is admissible.
Proof. If D is irreducible, then we are done. If not, there exists an i, such that D i
is admissible. If D i−1 or D i+1 is admissible, then by Proposition 5.3.3, all the D r are
admissible. So, it is enough to consider the case where neither D i−1 nor D i+1 is admissible
50

(and D i is admissible). We have:
r=i−1
∑
β 1 + β 2 + · · · + β i−1 < − v p (α r ),
r=1
r=i
∑
β 1 + β 2 + · · · + β i = − v p (α r ),
r=1
r=i+1
∑
β 1 + β 2 + · · · + β i+1 < − v p (α r ).
r=1
(5.3.4a)
(5.3.4b)
(5.3.4c)
Subtracting (5.3.4b) from (5.3.4a), we get −β i < v p (α i ). Subtracting (5.3.4b) from
(5.3.4c), we get β i+1 < −v p (α i+1 ) = −v p (α i ) + 1. Adding these inequalities, we obtain
β i+1 − β i < 1. But this is a contradiction, since β i+1 > β i . This proves the theorem.
Definition 5.3.6. Say π is p-ordinary, if β 1 + v p (α) = 0.
Note that if π is p-ordinary, then D 1 is admissible, so the flag D 1 ⊂ D 2 ⊂ · · · ⊂ D n
is an admissible flag by Theorem 5.3.5 (an easy check shows that if π is p-ordinary then
Assumption 4.4.6 holds automatically). Applying the above discussion to the local Galois
representation ρ π,p | Gp , we obtain:
Theorem 5.3.7 (Unramified twist of Steinberg). Say π is a cuspidal automorphic representation
with infinitesimal character given by the integers −β 1 > · · · > −β n . Suppose
that π p is an unramified twist of the Steinberg representation, i.e., WD(ρ π,p | Gp ) ∼
χ ⊗ Sp(n), where χ is the unramified character mapping arithmetic Frobenius to α. If π
is ordinary at p (i.e., v p (α) = −β 1 ), then the β i are necessarily consecutive integers and
ρ π,p | Gp ∼
⎛
α
λ( ) · χ −β ⎞
1
p vp(α) cyc,p ∗ · · · ∗
α
0 λ( ) · χ −β 1−1
p
⎜
vp(α) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ ,
α
0 0 0 λ( ) · χ −β 1−(n−1)
p vp(α) cyc,p
α
where λ( ) is an unramified character taking arithmetic Frobenius to , and in
p vp(α) p vp(α)
particular, ρ π,p | Gp is ordinary. If π is not p-ordinary, and Assumption 4.4.6 holds, then
ρ π,p | Gp is irreducible.
Proof. By strict compatibility, D is the filtered (ϕ, N, Q p , E)-module attached to ρ π,p | Gp .
If π is p-ordinary, then we are done and the characters on the diagonal are determined
by Corollary 4.2.11.
If π is not p-ordinary, then D is irreducible. Indeed, if D is reducible, then by Theorem
5.3.5, all D i , and in particular D 1 , are admissible, so π is p-ordinary.
5.3.2 Ramified twist of Steinberg
Theorem 5.3.8 (Ramified twist of Steinberg). Let the notation and hypotheses be as in
Theorem 5.3.7, except that this time assume that WD(ρ π,p | Gp ) ∼ χ⊗Sp(n), where χ is an
α
51

arbitrary, possibly ramified, character. Write χ = χ 0 · χ ′ where χ 0 is the ramified part of
χ, and χ ′ is an unramified character taking arithmetic Frobenius to α. If π is p-ordinary
(β 1 = −v p (α)), then the β i are consecutive integers and ρ π,p | Gp ∼
⎛
α
χ 0 · λ( ) · χ −β ⎞
1
p vp(α) cyc,p ∗ · · · ∗
α
0 χ 0 · λ( ) · χ −β 1−1
p
⎜
vp(α) cyc,p · · · ∗
⎝ 0 0 · · · ∗
⎟
⎠ .
α
0 0 0 χ 0 · λ( ) · χ −β 1−(n−1)
p vp(α) cyc,p
If π is not p-ordinary, and Assumption 4.4.6 holds, then ρ π,p | Gp
is irreducible.
Proof. Let F be a totally ramified abelian (cyclotomic) extension of Q p such that χ 0 | IF =
1. Then the reducibility of ρ π,p | GF over F can be shown exactly as in Theorem 5.3.7,
and the theorem over Q p follows using the descent data of the underlying filtered module.
If π is not p-ordinary, then by arguments similar those used in proving Theorem 5.3.7,
ρ π,p | GF is irreducible, so that ρ π,p | Gp is also irreducible.
5.4 Supercuspidal ⊗ Steinberg
We now turn to the case where the Weil-Deligne representation attached to π p is indecomposable.
We assume that WD(ρ π,p | Gp ) is Frobenius semisimple. Thus, it is of the
form τ ⊗ Sp(n), where τ is an irreducible m-dimensional representation corresponding to
a supercuspidal representation of GL m , for m ≥ 1.
We classify the (ϕ, N, F, E)-submodules of D, the crystal attached to the local representation
ρ π,p | Gp , where WD(ρ π,p | Gp ) = τ⊗Sp(n), for m ≥ 1 and n ≥ 1. This classification
will be used later to study the structure of ρ π,p | Gp , taking the filtration on D into account.
Recall that there is a equivalence of categories between (ϕ, N)-modules with coefficients
and descent data, and Weil-Deligne representations [BS07, Prop. 4.1]. Write D τ ,
respectively D Sp(n) , for the (ϕ, N)-modules corresponding to τ, respectively Sp(n), etc.
The main result of the first few subsections is:
Theorem 5.4.1. All the (ϕ, N, F, E)-submodules of D = D τ ⊗ D Sp(n) are of the form
D τ ⊗ D Sp(r) , for some 1 ≤ r ≤ n.
We prove the theorem in stages. However, the reader may turn straight to the §5.4.3,
where the general case is treated. Note that the case m = 1 was treated in the previous
section (twist of Steinberg), and the case n = 1 is vacuously true. The next simplest case
is when m = 2 and n = 2, and we start with this case in the next subsection.
The following lemma will be useful in our analysis throughout.
Lemma 5.4.2. The theory of Jordan canonical forms can be extended to nilpotent operators
on free of finite rank (F 0 ⊗ E)-modules, and we call the number of blocks in the
Jordan decomposition of the monodromy operator N as the ‘index’ of N.
Proof. One simply extends the usual theory of Jordan canonical forms on each projection
under (4.2.2) to modules over F 0 ⊗ E-modules.
52

5.4.1 m = 2 and n = 2
We start with the case when τ corresponds to a supercuspidal representation of GL 2 (Q p )
whose Weil-Deligne representation is the 2-dimensional representation obtained by inducing
a character from a quadratic extension K of Q p , and n = 2. The methods used in
this case will be used to deal with more general cases in subsequent subsections.
We also assume, for simplicity, that the quadratic extension of Q p mentioned above is
K = Q p 2, the unramified quadratic extension of Q p . By abusing language a bit we shall
say that τ corresponds to an unramified supercuspidal representation. The argument in
the ramified supercuspidal case (i.e., K/Q p ramified quadratic) was also worked out in
detail, but is excluded here for the sake of brevity.
Thus we assume that there is a character χ of W p 2, the Weil group of Q p 2, which does
not extend to W p (i.e., χ ≠ χ σ on W p 2, where 1 ≠ σ ∈ Gal(Q p 2/Q p )) such that
τ| Ip ≃ Ind Wp
W p 2 χ| I p
≃ χ| Ip ⊕ χ σ | Ip .
The corresponding (ϕ, N)-module in this case can be written down quite explicitly (cf.
[GM09, §3.3]). Briefly there is an abelian field F (depending on χ), generated by σ (a
lift to F of the automorphism σ above) and elements g of the inertia group I(F/K) of
Gal(F/K), and an element t ∈ E such that D τ = 〈e 1 , e 2 〉, with N = 0, and satisfies:
D τ = D unr−sc [a : b] =
⎧
⎪⎨
⎪⎩
ϕ(e 1 ) = 1 √
t
e 1 , ϕ(e 2 ) = 1 √
t
e 2 ,
σ(e 1 ) = e 2 , σ(e 2 ) = e 1 ,
g(e 1 ) = (1 ⊗ χ(g)) e 1 ,
g(e 2 ) = (1 ⊗ χ σ (g)) e 2 .
We do not write down the Hodge filtration since we do not need it here.
For the second factor, we recall that the module D Sp(2) has a basis f 1 , f 2 with properties
described at the start of §5.3.1. Then D = D τ ⊗ D Sp(2) has a basis of 4 vectors
{v i } 4 i=1 defined as follows v 1 = e 1 ⊗ f 2 v 2 = e 1 ⊗ f 1
v 3 = e 2 ⊗ f 2 v 4 = e 2 ⊗ f 1
Observe that on D, the monodromy operator N = 1 ⊗ N + N ⊗ 1 = 1 ⊗ N has kernel of
rank 2, and this space is generated by the vectors v 2 and v 4 .
For notational simplicity, write z for 1 ⊗ z, if z ∈ E and note that for z = ∑ i z i ⊗ e i ∈
F 0 ⊗ E, σ(z) = ∑ i σ(z i) ⊗ e i . Then the induced action on D, the tensor product of the
two (ϕ, N)-modules D τ and D Sp(2) , is summarized in the following table:
ϕ
v 1 = e 1 ⊗ f 2 v 2 = e 1 ⊗ f 1 v 3 = e 2 ⊗ f 2 v 4 = e 2 ⊗ f 1
√
p
t
v 1 √t 1
v 2 √t
p
v 3 √t 1
v 4
N v 2 0 v 4 0
σ v 3 v 4 v 1 v 2
g χ(g)v 1 χ(g)v 2 χ σ (g)v 3 χ σ (g)v 4
Lemma 5.4.3. There are no rank 1 (ϕ, N, F, E)-submodules of D.
53

Proof. Let 〈v〉 be a free module of rank-1 admissible (ϕ, N, F, E)-submodule of D. Write
v = av 1 + bv 2 + cv 3 + dv 4 where a, b, c, d ∈ (F 0 ⊗ Qp E). Since the rank of 〈v〉 is 1, we have
Nv is zero. From the above table, it is easy to see that v has to be equal to bv 2 + dv 4 , for
some b, d. Assume that ϕ(v) = α ϕ v. Since ϕ is bijective, α ϕ is a unit. We compute:
α ϕ (bv 2 + dv 4 ) = α ϕ v (1)
= ϕv (2)
= σ(b) 1 √
t
v 2 + σ(d) 1 √
t
v 4
(equality (1) follows from the fact that 〈v〉 is ϕ-stable and equality (2) from the table).
By comparing coefficients, we have
α ϕ
√
tb = σ(b), αϕ
√
td = σ(d). (5.4.1)
Lemma 5.4.4. Suppose x is a non-zero and a non-unit element of F 0 ⊗ E. Then σ(x) ≠
cx, for every c ∈ F 0 ⊗ E.
From the above relations for b (resp., d) and by lemma above, we conclude that b
(resp., d) is either zero or a unit.
Suppose that σ(v) = c σ v, for some c σ ∈ (F 0 ⊗ E), with c 2m
σ = 1 (this m is the m of
[GM09, §3.3], and not the m of this paper which is presently 2). Then
c σ (bv 2 + dv 4 ) = c σ v (1)
= σv (2)
= σ(b)v 4 + σ(d)v 2
(equality (1) follows from the fact that 〈v〉 is σ-stable and equality (2) from the table).
By comparing coefficients, we have
c σ b = σ(d), c σ d = σ(b). (5.4.2)
From these relations, we can conclude that b is zero if and only if d is zero. Since v is
non-zero, we have that both b and d are both units.
For every g ∈ I(F/K), we have
c g (bv 2 + dv 4 ) = c g v (1)
= g · v (2)
= χ(g)bv 2 + χ σ (g)dv 4
(equality (1) follows from the fact that 〈v〉 is I(F/K)-stable and equality (2) from the
table). Now, by comparing coefficients, and using the fact that b and d are units, we have
c g = χ(g),
c g = χ σ (g).
Hence χ(g) = χ σ (g), for every g ∈ I(F/K). This is a contradiction.
Remark 5.4.5. The above argument shows that there are no (ϕ, Gal(F/Q p ))-stable submodules
of rank 1 of either 〈v 2 , v 4 〉 or, as one can show similarly, 〈v 1 , v 3 〉. There is also a
simpler proof of this Lemma which avoids working with the above explicit manipulations
which works for general m and n (cf. the proof of Lemma 5.4.11).
Lemma 5.4.6. The only rank 2 (ϕ, N, F, E)-submodule of D is 〈v 2 , v 4 〉 = D τ ⊗ D Sp(1) .
54

Proof. Let D ′ be a filtered module of rank 2. Suppose the index of N = 1, i.e., D ′ =
〈w 1 , N(w 1 )〉, such that N 2 (w 1 ) = 0. It is easy to see that 〈N(w 1 )〉 is a rank 1 (ϕ, N, F, E)-
submodule of D, a contradiction by Lemma 5.4.3. Therefore, the index of N ≠ 1.
Suppose the index of N = 2, i.e., D ′ = 〈w 1 , w 2 〉 such that N(w 1 ) = N(w 2 ) = 0. We
know that the kernel of N is 〈v 2 , v 4 〉. Therefore, we have that 〈w 1 , w 2 〉 ⊆ 〈v 2 , v 4 〉. But
both are free modules of the same rank, hence the equality holds.
Lemma 5.4.7. There are no rank 3 (ϕ, N, F, E)-submodules of D.
Proof. Let D ′ be a filtered module of D of rank 3. The index of nilpotency of N cannot
be 3, since the kernel of N has rank 2.
Suppose the index of nilpotency of N is 2, that is, there exists a basis, say w 1 , w 2 ,
and N(w 2 ), of D ′ such that N(w 1 ), N 2 (w 2 ) are both zero. Clearly 〈w 1 , N(w 2 )〉 = 〈v 2 , v 4 〉.
Thus there is a vector w 2 ′ = a 1v 1 + a 3 v 3 such that D ′ = 〈w 2 ′ , v 2, v 4 〉. But 〈w 2 ′ 〉 is stable
by ϕ. Indeed ϕ acts by a scalar on w 2 ′ since, a priori, ϕw′ 2 is a linear combination of
w 2 ′ , v 2 and v 4 , but the v 2 and v 4 components do not appear since ϕ preserves the space
〈v 1 , v 3 〉. Similarly 〈w 2 ′ 〉 is also Gal(F/Q p)-stable. But then 〈w 2 ′ 〉 ⊂ 〈v 1, v 3 〉 is a rank 1
module stable by ϕ and Gal(F/Q p ), which is not possible (cf. Remark 5.4.5).
Finally, suppose the index of nilpotency of N is 1. But this case does not arise since
the index of nilpotency of N on D is 2.
Proof of Theorem 5.4.1 when m = 2 and n = 2
This follows immediately from Lemmas 5.4.3, 5.4.6, and 5.4.7, when τ is an unramified
supercuspidal representation of dimension m = 2, and n = 2. As we remarked earlier, the
case when τ is a ramified supercuspidal representation is proved in a very similar manner
using the notation in [GM09, §3.4] (the only difference in the computations are the role
of σ is now played by the automorphism ι there).
We mention some immediate corollaries of Theorem 5.4.1 in the present case. Let
π be an automorphic representation of GL 4 (A Q ) with infinitesimal character consisting
of distinct integers −β 4 < · · · < −β 1 . Let ρ = ρ π,p | Gp be the corresponding (p, p)-
Galois representation. Suppose that WD(ρ) ∼ τ ⊗ Sp(2), where τ is a supercuspidal
representation of dimension 2, as above. Let D = D(ρ) be the corresponding admissible
filtered (ϕ, N, F, E)-module. Note that β 4 > β 3 > β 2 > β 1 are also the drops in the Hodge
filtration on D F .
Corollary 5.4.8. With notation as above, the crystal D is irreducible if and only if
〈v 2 , v 4 〉 is not an admissible (ϕ, N, F, E)-submodule of D.
Proof. By Lemmas 5.4.3 and 5.4.7, there are no rank 1 and rank 3 filtered submodules
of D. By Lemma 5.4.6, there exists a unique rank 2 (ϕ, N, F, E)-submodule of D, which
is 〈v 2 , v 4 〉. Hence the admissibility of 〈v 2 , v 4 〉 is equivalent to the reducibility of D.
Corollary 5.4.9. The crystal D is irreducible if no two of the four β i add up to −v p (t).
In particular, ρ is irreducible in this case.
Proof. By Corollary 5.4.8, D is irreducible if and only if D ′ = 〈v 2 , v 4 〉 is not an admissible
submodule. The submodule D ′ is not admissible if no two of the four β i add up to t N (D ′ )
which is −v p (t), from the table above.
55

5.4.2 τ unramified supercuspidal of dim m ≥ 2 and n ≥ 2
We now prove Theorem 5.4.1 for general m and n, assuming τ is an ‘unramified supercuspidal
representation’. Let us explain this terminology. We assume that τ is an induced
representation of dimension m, i.e., τ ≃ Ind Wp
W K
χ, where K is a p-adic field such that
[K : Q p ] = m, and χ is a character of W K . This is known to always hold if (p, m) = 1 or
p > m. For simplicity, we shall assume that K is the unique unramified extension of Q p ,
namely Q p m. We refer to τ in this case as an unramified supercuspidal representation.
Following the methods of [GM09, §3.3], we explicitly write down the crystal D = D τ
whose underlying Weil-Deligne representation is an unramified supercuspidal representation
τ of dimension m. This is done in the next few subsections. The arguments are
similar to those given in [GM09, §3.3], with some minor modifications. We outline the
steps now.
Let σ be the generator of Gal(Q p m/Q p ) and let I p m denote the inertia subgroup of
Q p m. Then
τ| Ip=Ip m ≃ (Ind Wp
W χ)| p m I p m ≃ ⊕ m i=1χ σi | Ip m .
Since τ is irreducible, by Mackey’s criterion, we have that χ ≠ χ σi , for all i, on W p m
also on I p m. Moreover, we have that χ σj ≠ χ σi , for any i ≠ j.
and
Description of Gal(F/Q p )
First, we need to construct a finite extension F over K such that τ| IF is trivial and which
simultaneously has the property that F/Q p is Galois. The construction of an explicit such
field F is given in [GM09, §3.3.1], in the case K = Q p 2 using Lubin-Tate theory. The
structure of Gal(F/Q p ) is described in the same place. The case K = Q p m may be treated
in a very similar manner. Write d for what was called in m in [GM09, §3.3.1], since m
already has meaning here. Then F may be chosen such that Gal(F/Q p ) is the semi-direct
product of a cyclic group 〈σ〉 with σ md = 1, with the product of the cyclic groups ∆ = 〈δ〉,
with δ pm−1 = 1, and Γ = ∏ m
i=1 〈γ i〉, with each γ pn
i
= 1, for some n. Moreover, the maximal
unramified extension F 0 of F is F 0 = Q md
p and Gal(F 0 /Q p ) = 〈˜σ〉 = Z/md, such that ˜σ| K
is the generator of Gal(K/Q p ). By abuse of notation, we denote ˜σ by σ itself.
Description of the Galois action
Recall that D is a free module of rank m over F 0 ⊗ Qp E . Let D i = D ⊗ F0 ⊗E,σi E, for
i = 0, 1, . . . , md−1, be the component of D corresponding to σ i . Each D i is a Weil-Deligne
representation with an action of W p .
By the definition of the Weil-Deligne representation, the action of I p matches with the
action of the inertia subgroup of Gal(F/Q p ), namely ∆×Γ. The restriction of χ to I p can
be written as χ| Ip = ωm r ∏ m
i=1 χ i, where ω m is the fundamental character of level m, r ≥ 1,
and χ i is the character of Γ which takes γ i to a p n -th root of unity ζ i , for i = 1, . . . , m.
We see that each D i has a basis v i,1 , v i,2 , . . . , v i,m such that if i ≡ k (mod m), for some
0 ≤ k ≤ m − 1, then
g · v i,j = χ σj−k−1 (g)v i,j , (5.4.3)
56

for all j = 1, . . . , m. Since σ takes D i to D i+1 , using (5.4.3), we may assume that
Description of action of ϕ
σ(v i,j ) = v i+1,j ∀i, j.
The operator ϕ acts in a cyclic manner as well, taking D i to D i+1 . Since ϕ commutes
with the action of inertia, we see that
ϕ(v i,j ) = c j v i+1,j+1 ,
for some c j , for all 1 ≤ j ≤ m. Observe that c j ’s does not depend on i, since ϕ commutes
with σ.
1
For all 1 ≤ k ≤ m, define,
t k
:= ∏ k
j=1 c j and 1 t 0
= 1. Replace the extension E with a
finite extension, again denoted by E, so that it contains all m-th roots of all c j . Let m√ c j
denote a particular m-th root, for each j.
We now write down a basis of D, say {e i } m i=1 , such that ϕ(e i) = 1 m √ e
t i . First, we
m
shall define e 1 and the other e i ’s are defined by e i = σ i−1 e 1 . The vector e 1 is given by
e 1 =
md−1
∑
j=0
j≡j 0 (mod m), 0≤j 0 ≤m−1
(t m ) j 0
m
t j0
v j,j+1 .
Here we use the obvious convention that if j is such that j ≡ j 0 (mod m), with 1 ≤ j 0 ≤ m,
then v i,j := v i,j0 . A small computation shows that ϕ(e 1 ) = 1 m √ t m
e 1 . Since ϕ commutes
with σ, we have ϕ(e i ) = 1 m √ e
t i , for all 1 ≤ i ≤ m. We obtain the D τ is a free rank m
m
module over F 0 ⊗ E with basis e i , i = 1, . . . , m such that
⎧
1
ϕ(e i ) = m √ e
t i ,
m
⎪⎨
N(e i ) = 0,
D τ =
(5.4.4)
σ(e i ) = e i+1 ,
⎪⎩
g(e i ) = (1 ⊗ χ σi−1 (g))(e i ), g ∈ I(F/K),
for all 1 ≤ i ≤ m.
When m = 2, this (ϕ, N)-module is exactly the one given in [GM09, §3.3] (though the
e i used here differ by a scalar from the e i used there).
Description of the filtration
For the sake of completeness, let us make some brief comments about the filtration on
D τ , even though we shall not need to use the filtration later.
Let D be a filtered (ϕ, N, F, E)-module and write D F = F ⊗ F0 D. It is known that
every Galois stable line in D F is generated by a Galois stable vector v (cf. [GM09, Lem.
3.1]). The proof uses the fact H 1 (Gal(F/Q p ), (F ⊗ Qp E) × ) = 0. In fact, for any n ≥ 1:
∏
H 1 (Gal(F/Q p ), GL n (F ⊗ Qp E)) = H 1 (Gal(F/Q p ), GL n (E)),
F ↩→E
=
(1) H1 (Gal(F/Q p ), Ind Gal(F/Qp)
{e}
GL n (E)),
=
(2) H1 ({e}, GL n (E)) = {0},
57

(where (1) follows from the permutation action of Gal(F/Q p ) on
follows from Shapiro’s lemma). Now, we can prove:
∏
F ↩→E
GL n (E) and (2)
Lemma 5.4.10. Every Gal(F/Q p )-stable submodule of D F has a basis consisting of Galois
invariant vectors.
Proof. Let D ′ be a Galois stable submodule of D F . By [Sav05, Lem. 2.1], we have that
any F ⊗ Qp E-submodule of a filtered module with descent data is free. Hence D ′ is a
free module of finite rank, say r. If {v 1 , v 2 , . . . , v r } is a basis of D ′ , then for every g ∈
Gal(F/Q p ), we have g ·(v 1 , v 2 , · · · , v n ) t = c g (v 1 , v 2 , · · · , v r ) t , for some c g ∈ GL n (F ⊗ Qp E).
Moreover, c g is 1-cocycle, i.e., c g ∈ Z 1 (Gal(F/Q p ), GL n (F ⊗ Qp E)). By the vanishing result
above, c g is coboundary, hence c g = cg(c) −1 , for some c ∈ GL n (F ⊗ Qp E). Replacing the
basis (v 1 , v 2 , · · · , v r ) t with c · (v 1 , v 2 , · · · , v r ) t , we may assume that c g = 1 and that each
vector in {v 1 , v 2 , . . . , v r } is invariant under Gal(F/Q p ).
In particular each step Fil i (D F ) in the filtration on D F is spanned by Gal(F/Q p )-
invariant vectors. In [GM09, §3.3.4] the Hodge filtration on D F was written down explicitly
when D = D τ and τ is a 2-dimensional unramified supercuspidal representation.
Presumably this can be done also when τ has dimension m ≥ 2, but we refrain from
pursuing this.
Using the explicit description of the crystal above, we can prove Theorem 5.4.1 in the
case above, and for brevity, we skip the proof and prove it in the general case directly.
5.4.3 General case: m ≥ 2 and n ≥ 2
Now, we shall prove Theorem 5.4.1 in general. Thus, we show that the only (ϕ, N, F, E)-
submodules of D = D τ ⊗ D Sp(n) , for any irreducible representation τ of W p dimension
m ≥ 2, and n ≥ 2, are of the form D τ ⊗ D Sp(r) , for some 1 ≤ r ≤ n. The proof uses ideas
introduced for the special cases proved so far.
Recall that the module D Sp(n) has a basis {f n , f n−1 , . . . , f 1 }, with properties as in
§5.3.1, and say that D τ has a basis {e 1 , e 2 , . . . , e m } over F 0 ⊗ E. Let {v i } mn
i=1 denote the
basis of D = D τ ⊗ D Sp(r) defined by the following table:
v 1 = e 1 ⊗ f n v 2 = e 1 ⊗ f n−1 · · · v n = e 1 ⊗ f 1
v n+1 = e 2 ⊗ f n v n+2 = e 2 ⊗ f n−1 · · · v 2n = e 2 ⊗ f 1
.
. · · ·
v in+1 = e i+1 ⊗ f n v in+2 = e i+1 ⊗ f n−1 · · · v in = e i+1 ⊗ f 1
.
. · · ·
v (m−1)n+1 = e m ⊗ f n v (m−1)n+2 = e m ⊗ f n−1 · · · v mn = e m ⊗ f 1
.
.
Lemma 5.4.11. There are no rank r (ϕ, N, F, E)-submodules of D on which N acts
trivially, for 1 ≤ r ≤ m − 1.
Proof. Suppose there exists such a module, say ˜D, of rank r < m. Since N acts trivially
on ˜D, we have ˜D ⊆ 〈v n , v 2n , . . . , v mn 〉 = D τ ⊗ D Sp(1) ≃ D τ . But τ is irreducible, so D τ is
irreducible by Lemma 4.2.14, a contradiction.
58

Corollary 5.4.12. The index of N on a (ϕ, N, F, E)-submodules of D is m.
Proof of Theorem 5.4.1. Let D ′ be a (ϕ, N, F, E)-submodule of D = D τ ⊗ D Sp(n) . From
the corollary above, there are m blocks in the Jordan canonical form of N on D ′ . Without
loss of generality assume that the blocks have sizes r 1 ≤ r 2 ≤ · · · ≤ r m with ∑ m
i=1 r i =
rank D ′ . Suppose w 1 , . . . , w m are the corresponding basis vectors in D ′ such that the
order of nilpotency of N on w i is r i , so that the {N j (w i )} i,j form a basis of D ′ . If all the
r i are equal to say r, then the usual argument shows D ′ = D τ ⊗ D Sp(r) . We show that
this is indeed the case.
Suppose towards a contradiction that r i ≠ r i+1 for some 1 ≤ i < m. For 1 ≤ i ≤ n,
let D i be span of the vectors in the last i columns in the table at the start of this section.
Observe that D i = D τ ⊗ Ker(N i ) = D τ ⊗ D Sp(i) .
Now, arrange the basis vectors of D ′ , i.e., the N j w k , in such a way that the last
column in the table contains the basis vectors on which N acts trivially and the last but
one column consists of basis vector which precisely spans Ker(N 2 ) on D ′ , and so on and
so forth. With respect to this arrangement, denote the span of the vectors in the last i
columns as A i . Since r i ≠ r i+1 , the rank of the space A ri +1/A ri is less than m. Moreover,
A ri +1/A ri is a subspace of D ri +1/D ri , i.e., there is an inclusion of (ϕ, N, F, E)-modules
A ri +1/A ri ↩→ D ri +1/D ri .
Now
D ri +1/D ri = (D τ ⊗ D Sp(ri +1))/(D τ ⊗ D Sp(ri ))
≃ D τ ⊗ (D Sp(ri +1)/D Sp(ri ))
≃ D τ ⊗ D Sp(1)
≃ D τ .
All the isomorphisms above are isomorphism of (ϕ, N, F, E)-modules over F 0 ⊗ E. By
Lemma 5.4.11, the above inclusion is not possible! Hence all the r i are indeed equal. This
finishes the proof of Theorem 5.4.1.
Filtration on D = D τ ⊗ D Sp(n)
We finally can apply the discussion above to write down the structure of the (p, p)-
representation attached to an automorphic representation of GL mn (A Q ).
We start with some remarks. Suppose D 1 and D 2 are two admissible filtered modules.
It is well-known (cf. [Tot96]) that the tensor product D 1 ⊗ D 2 is also admissible. The
difficulty in proving this lies in the fact that one does not have much information about
the structure of the (ϕ, N)-submodules of the tensor product. If they are of the form
D ′ ⊗ D ′′ , where D ′ and D ′′ are admissible (ϕ, N)-submodules of D 1 and D 2 respectively,
then one could use (4.2.1) to prove that D ′ ⊗ D ′′ is also admissible. But not all the
submodules of D 1 ⊗ D 2 are of this form.
However in the previous section we have just shown (cf. Theorem 5.4.1), that for
D = D τ ⊗ D Sp(n) , all the (ϕ, N, F, E)-submodules of D are of the form D τ ⊗ D Sp(r) , for
some 1 ≤ r ≤ n. This fact allows us to study the crystal D and its submodules, once we
introduce the Hodge filtration.
59

Filtration in general position
Assume that the Hodge filtration on D is in general position with respect to the Newton
filtration (cf. Assumption 4.4.6). Let m be the rank of D τ . Let {β i,j } i=n,j=m
i=1,j=1
be the
jumps in the Hodge filtration with β i1 ,j 1
> β i2 ,j 2
, if i 1 > i 2 , or if i 1 = i 2 and j 1 > j 2 . Thus
β n,m > β n,m−1 > · · · > β n,1 > β n−1,m > · · · > β 1,m > · · · > β 1,1 .
Define, for every 1 ≤ k ≤ n,
and
b k =
j=m
∑
j=1
β k,j ,
a k = t N (D τ ⊗ D Sp(k) ) − t N (D τ ⊗ D Sp(k−1) ) = t N (D τ ) + m(k − 1),
where the last equality follows from (4.2.1a). Clearly, we have that
b n > · · · > b i+2 > b i+1 > b i > · · · > b 1 ,
a n > · · · > a i+2 > a i+1 > a i > · · · > a 1 .
(5.4.5)
Observe that b i+1 −b i ≥ m 2 and a i+1 −a i = m, for every 1 ≤ i ≤ n. Since D is admissible,
the submodule D τ ⊗ D Sp(i) of D is admissible if and only if ∑ i
k=1 b k = ∑ i
k=1 a k.
The arguments below are similar to the ones used when analyzing the Steinberg case.
We start with an analog of Lemma 5.3.2.
Lemma 5.4.13. Let {a i } n i=1 be an increasing sequence of integers, such that a i+1−a i = m,
for every i and for some fixed natural number m. Let {b i } n i=1 be an increasing sequence
of integers, such that b i+1 − b i ≥ m 2 , for every i. Suppose that ∑ i a i = ∑ i b i. If a n = b n
or a 1 = b 1 , then m = 1 and hence a i = b i , for all i.
Theorem 5.4.14. If D τ ⊗D Sp(i) and D τ ⊗D Sp(i+1) are admissible submodules of D, then
m = 1, in which case all the D τ ⊗ D Sp(i) , for 1 ≤ i ≤ n, are admissible.
Proof. Since D τ ⊗ D Sp(i) and D τ ⊗ D Sp(i+1) are admissible, we have:
b 1 + b 2 + · · · + b i =
i∑
a r ,
r=1
i+1
∑
b 1 + b 2 + · · · + b i+1 = a r .
r=1
(5.4.6)
Clearly, we have b i+1 = a i+1 . By (5.4.5), (5.4.6), and by Lemma 5.4.13 we have m = 1
and a i = b i , for all 1 ≤ i ≤ n. This shows that all the D τ ⊗ D Sp(i) are admissible.
Theorem 5.4.15. Let D = D τ ⊗ D Sp(n) and assume that the Hodge filtration on D is in
general position (cf. Assumption 4.4.6). Then either D is irreducible or D is reducible,
in which case m = 1 and the (ϕ, N, F, E)-submodules D τ ⊗ D Sp(i) , for 1 ≤ i ≤ n are all
admissible.
60

Proof. Let D i = D τ ⊗ D Sp(i) , for 1 ≤ i ≤ n. If D is irreducible then we are done. If
not, by Theorem 5.4.1, there exists an 1 ≤ i ≤ n such that D i is admissible. If D i−1 or
D i+1 is also admissible, then by the above theorem m = 1 and hence all the (ϕ, N, F, E)-
submodules of D are admissible. So, assume D i−1 and D i+1 are not admissible, but D i
is admissible. We shall show that this is not possible. Indeed, we have:
b 1 + b 2 + · · · + b i−1 <
r=i−1
∑
r=1
r=i
a r ,
∑
b 1 + b 2 + · · · + b i = a r ,
b 1 + b 2 + · · · + b i+1 <
r=1
r=i+1
∑
r=1
a r .
(5.4.7a)
(5.4.7b)
(5.4.7c)
Subtracting (5.4.7b) from (5.4.7a), we get −b i < −a i . Subtracting (5.4.7b) from (5.4.7c),
we get b i+1 < a i+1 . Adding these two inequalities, we get b i+1 − b i < a i+1 − a i = m. But
this is a contradiction, since b i+1 − b i ≥ m.
For emphasis we state separately the following corollary:
Corollary 5.4.16. With assumptions as above, for any m ≥ 2, the crystal D = D τ ⊗
D Sp(n) is irreducible.
Definition 5.4.17. Say π is ordinary at p if a 1 = b 1 , i.e., t N (D τ ) = ∑ m
j=1 β 1,j.
This condition implies m = 1, and this definition then coincides with Definition 5.3.6.
Applying the above discussion to the local (p, p)-representation in a strictly compatible
system, we obtain:
Theorem 5.4.18 (Indecomposable case). Say π is a cuspidal automorphic representation
with infinitesimal character consisting of distinct integers. Suppose that
WD(ρ π,p | Gp ) ∼ τ m ⊗ Sp(n),
where τ m is an irreducible representation of W p of dimension m ≥ 1, and n ≥ 1. Assume
that Assumption 4.4.6 holds.
• If π is ordinary at p, then ρ π,p | Gp is reducible, in which case m = 1 and τ 1 is a
character and ρ π,p | Gp is (quasi)-ordinary as in Theorems 5.3.7 and 5.3.8.
• If π is not ordinary at p, then ρ π,p | Gp
is irreducible.
Tensor product filtration
One might wonder what happens if the filtration on D is not necessarily in general position.
As an example, here we consider just one case arising from the so called tensor
product filtration.
Assume that D τ and D Sp(n) are the usual filtered (ϕ, N, F, E)-modules and equip
D τ ⊗ D Sp(n) with the tensor product filtration. By the formulas in Lemma 4.2.7 one can
prove:
61

Lemma 5.4.19. Suppose that D = D τ ⊗ D Sp(n) has the tensor product filtration. Fix
1 ≤ r ≤ n. Then D τ ⊗ D Sp(r) is an admissible submodule of D if and only if D Sp(r) is an
admissible submodule of D Sp(n) .
We recall that if the filtration on D Sp(n) is in general position (as in Assumption 4.4.6),
then we have shown that furthermore D Sp(r) is an admissible submodule of D Sp(n) if and
only if D Sp(1) is an admissible submodule.
Remark 5.4.20. The lemma can be used to give an example where the tensor product
filtration on D is not in general position (i.e., does not satisfy Assumption 4.4.6). Suppose
τ is an irreducible representation of dimension m = 2 and D Sp(2) has weight 2 (cf. [GM09,
§3.1]). Note 〈f 1 〉 is an admissible submodule of D Sp(2) . Hence, by the lemma, D τ ⊗ 〈f 1 〉
is an admissible submodule of D τ ⊗ D Sp(2) . If the tensor product filtration satisfies
Assumption 4.4.6, then the admissibility of D τ ⊗ 〈f 1 〉 would contradict Theorem 5.4.15,
since m = 2.
In any case, we have the following application to local Galois representations.
Proposition 5.4.21. Suppose that ρ π,p | Gp ∼ ρ τ ⊗ ρ Sp(n) is a tensor product of two (p, p)-
representations, with underlying Weil-Deligne representations τ and Sp(n) respectively.
If ρ Sp(n) is irreducible, then so is ρ π,p | Gp .
5.5 General Weil-Deligne representations
So far, we have studied the (p, p)-Galois representation attached to π p when the underlying
Weil-Deligne representation is indecomposable. We now make some remarks in the
general setting where the Weil-Deligne representation is a direct sum of indecomposable
representations.
5.5.1 Sum of twisted Steinberg
For simplicity we start with the case where the indecomposable pieces are twists of the
Steinberg representation by an unramified character. Thus we assume the underlying
Weil-Deligne representation is
D χ1 ⊗ D Sp(n1 ) ⊕ D χ2 ⊗ D Sp(n2 ) ⊕ · · · ⊕ D χr ⊗ D Sp(nr),
where n i ≥ 1 and χ i are unramified characters taking arithmetic Frobenius to α i . Let
χ i (ω) = α i where ω is a uniformizer of Q × p . Without loss of generality we may assume
that v p (α 1 ) ≥ v p (α 2 ) ≥ · · · ≥ v p (α r ).
Let n = ∑ r
i n i. Let {β i,j } r,n i
i=1,j=1
be the jumps in the Hodge filtration such that
β i1 ,j 1
> β i2 ,j 2
, if i 1 > i 2 or i 1 = i 2 and j 1 > j 2 . Thus
β r,nr > · · · > β r,1 > β r−1,nr−1 > · · · > β r−1,1 > · · · >
β 2,n2 > · · · > β 2,1 > β 1,n1 > · · · > β 1,1 .
(5.5.1)
62

Let D be a filtered (ϕ, N, Q p , E)-module with associated Weil-Deligne representation
as above. We now define a flag inside D as follows.
⎧
D χ1 ⊗ D Sp(i) if 1 ≤ i ≤ n 1 ,
⎪⎨ D χ1 ⊗ D Sp(n1 ) ⊕ D χ2 ⊗ D Sp(i−n1 ) if n 1 + 1 ≤ i ≤ n 1 + n 2 ,
D i =
.
⎪⎩
⊕ r−1
k=1 D χ i
⊗ D Sp(ni ) ⊕ D χr ⊗ D Sp(i−(n−nr)) if n − n r + 1 ≤ i ≤ n.
Clearly, D n is the full (ϕ, N)-module D. We now show that the above flag is admissible
if and only if π is ordinary at p in the following sense:
Definition 5.5.1. Say π is ordinary at p if β i,1 = −v p (α i ) for all 1 ≤ i ≤ r.
We remark that the notion of ordinariness extends the previous definitions given in
Definition 5.2.1, when all the n i = 1 and Definition 5.3.6, when r = 1 and m = 1. We
have:
Theorem 5.5.2. Assume that Assumption 4.4.6 holds. Then the flag {D i } is an admissible
flag in D (i.e., each D i is admissible) if and only if π is ordinary at p.
Proof. The ‘only if’ part is clear. Indeed if the Hodge filtration is in general position
then the jump in the filtration on D 1 will be the last number in (5.5.1), i.e., β 1,1 , and the
admissibility of D 1 shows that β 1,1 = −v p (α 1 ). Similarly, the jumps in the filtration on
D n1 (respectively D n1 +1) are the last n 1 numbers (respectively the last n 1 numbers along
with β 2,1 ) in (5.5.1) above, and clearly t N (D n1 +1) = t N (D n1 )−v p (α 2 ), so the admissibility
of D n1 and D n1 +1 together shows that β 2,1 = −v p (α 2 ), etc.
Let us prove the ‘if’ part. Since β 1,1 = −v p (α 1 ), D 1 is admissible. Since D 2 is a
(ϕ, N, Q p , E)-submodule of D we have that
β 1,1 + β 1,2 ≤ (1 − v p (α 1 )) + (−v p (α 1 ))
and hence β 1,2 ≤ 1 − v p (α 1 ) = 1 + β 1,1 . Thus β 1,2 − β 1,1 ≤ 1. But β 1,2 − β 1,1 ≥ 1, by
(5.5.1), hence equality holds, i.e., β 1,2 = (1 − v p (α 1 )). Therefore,
By a similar argument, we see that
β 1,1 + β 1,2 = (1 − v p (α 1 )) + (−v p (α 1 )).
∑n 1
j=1
∑n 1
β 1,j = (j − 1 − v p (α 1 )). (5.5.2)
j=1
This shows that D n1 is admissible. Since D n1 +1 is an (ϕ, N, F, E)-submodule of D, we
have that
∑n 1
∑n 1
β 1,j + β 2,1 ≤ (j − 1 − v p (α 1 )) + (−v p (α 2 )),
j=1
j=1
but this inequality is actually an equality, by (5.5.2), and since β 2,1 = −v p (α 2 ) by assumption.
This shows that D n1 +1 is also admissible. The admissibility of the other D i is
proved in a similar manner.
63

5.5.2 General ordinary case
We now assume that as a (ϕ, N, F, E)-module,
D = D τ1 ⊗ D Sp(n1 ) ⊕ D τ2 ⊗ D Sp(n2 ) ⊕ · · · ⊕ D τr ⊗ D Sp(nr),
where n i ∈ N and τ i ’s are irreducible representations of W p of degree m i . Without loss of
generality we may assume t N (D τ1 ) ≤ t N (D τ2 ) ≤ · · · ≤ t N (D τr ).
We now define a flag inside D τ1 ⊗ D Sp(n1 ) ⊕ · · · ⊕ D τr ⊗ D Sp(nr), and show that this
flag is admissible if and only if there is a relation between some numbers a i (depending
on Newton numbers) and b i (depending on Hodge numbers). More precisely, define the
flag {D i } in D by
⎧
D τ1 ⊗ D Sp(i) if 1 ≤ i ≤ n 1 ,
⎪⎨ D τ1 ⊗ D Sp(n1 ) ⊕ D τ2 ⊗ D Sp(i−n1 ) if n 1 + 1 ≤ i ≤ n 1 + n 2 ,
D i =
.
⎪⎩
⊕ r−1
k=1 D τ k
⊗ D Sp(nk ) ⊕ D τr ⊗ D Sp(i−(n−nr)) if n − n r + 1 ≤ i ≤ n.
We now define the numbers a i and b i . Let {β i,j } be the jumps in the Hodge filtration
associated to D such that β i1 ,j 1
> β i2 ,j 2
, if i 1 > i 2 or if i 1 = i 2 but j 1 > j 2 . Thus, in the
case r = 2, the jumps in the filtration are:
β m1 +m 2 ,n 2
> β m1 +m 2 −1,n 2
> · · · > β m1 +1,n 2
>
β m1 +m 2 ,n 2 −1 > β m1 +m 2 −1,n 2 −1 > · · · > β m1 +1,n 2 −1 > · · · >
β m1 +m 2 ,1 > β m1 +m 2 −1,1 > · · · > β m1 +1,1 >
β m1 ,n 1
> β m1 −1,n 1
> · · · > β 1,n1 >
β m1 ,n 1 −1 > β m1 −1,n 1 −1 > · · · > β 1,n1 −1 > · · · >
Define, for every 1 ≤ k ≤ n 1 ,
b k =
i=m
∑ 1
i=1
β m1 ,1 > β m1 −1,1 > · · · > β 1,1 .
β i,k ,
a k = t N (D τ1 ⊗ D Sp(k) ) − t N (D τ1 ⊗ D Sp(k−1) ) = t N (D τ1 ) + m 1 (k − 1).
Clearly, we have that
b n1 > b n1 −1 > · · · > b 2 > b 1 ,
a n1 > a n1 −1 > · · · > a 2 > a 1 .
Observe that b i+1 − b i ≥ m 2 1 and a i+1 − a i = m 1 , for 1 ≤ i < n 1 . Under Assumption 4.4.6,
the jumps in the induced Hodge filtration on D τ1 ⊗ D Sp(j) are β m1 ,j > · · · > β 1,1 , so that
D τ1 ⊗ D Sp(j) is an admissible submodule of D if and only if ∑ j
k=1 b k = ∑ j
k=1 a k.
Similarly define b k and a k for all 1 ≤ k ≤ n = ∑ r
i n i. For example, if n 1 +1 ≤ n 1 +k ≤
n 1 + n 2 , define
b n1 +k =
i=m
∑ 2
i=1
β m1 +i,k,
a n1 +k = t N (D τ2 ⊗ D Sp(k) ) − t N (D τ2 ⊗ D Sp(k−1) ) = t N (D τ2 ) + m 2 (k − 1).
64

Again, we have
b n1 +n 2
> b n1 +n 2 −1 > · · · > b n1 +2 > b n1 +1,
a n1 +n 2
> a n1 +n 2 −1 > · · · > a n1 +2 > a n1 +1,
and b i+1 − b i ≥ m 2 2 and a i+1 − a i = m 2 , for every n 1 + 1 ≤ i < n 1 + n 2 , etc.
Definition 5.5.3. Say π is ordinary at p if a P i−1
j=1 n j+1 = bP i−1
j=1 n , for 1 ≤ i ≤ r.
j+1
Note that this definition of ordinariness reduces to Definition 5.4.17 when r = 1, but
also to Definition 5.5.1 since it implies m i = 1, for 1 ≤ i ≤ r.
Theorem 5.5.4. Assume that Assumption 4.4.6 holds. Then, the flag {D i } is admissible
(i.e., each D i is an admissible submodule of D) if and only if π is ordinary at p.
Proof. We prove the ‘only if’ direction for r = 2, since the general case is similar, and only
notationally more cumbersome. Thus we have to show that if the flag {D i } is admissible,
then a 1 = b 1 and a n1 +1 = b n1 +1. The proof is an easy application of Lemma 5.4.13.
Indeed
• The admissibility of D 1 shows that a 1 = b 1 .
• The admissibility of D n1
Lemma 5.4.13).
and a 1 = b 1 shows m 1 = 1 and a i = b i for 1 ≤ i ≤ n 1 (by
• The admissibility of D n1 +1 and D n1 together shows a n1 +1 = b n1 +1.
• The admissibility of D n1 +n 2
and D n1 and a n1 +1 = b n1 +1 shows m 2 = 1 and a n1 +i =
b n1 +i for 1 ≤ i ≤ n 2 (by Lemma 5.4.13).
Since all m i = 1, the proof of the ‘if’ part of the theorem is exactly the same as the ‘if’
part of the proof of Theorem 5.5.2, noting b P i−1
j=1 n j+1 = β i,1 and and a P i−1
j=1 n j+1 = −v p(α i ),
for 1 ≤ i ≤ r.
Remark 5.5.5. The theorem does not tell us when D is irreducible, since there are a large
number of (ϕ, N, F, E)-submodules of D which are not part of the flag considered above.
For instance, it seems hard to determine the admissibility of the submodule D τi ⊗D Sp(ni ),
except when i = 1.
Translating the above theorem in terms of the (p, p)-representation, we obtain:
Theorem 5.5.6 (Decomposable case). Suppose π is a cuspidal automorphic representation
of GL N (A Q ) with infinitesimal character given by the integers −β 1 > · · · > −β N .
Suppose that N = ∑ r
i=1 m in i and
WD(ρ π,p | Gp ) ∼ ⊕ r i=1τ i ⊗ Sp(n i ),
where τ i is an irreducible representation of dimensions m i ≥ 1, and n i ≥ 1. If π is
ordinary at p, then m i = 1 for all i, the β i occur in r blocks of consecutive integers, of
lengths n i , for 1 ≤ i ≤ r, and
⎛
⎞
ρ n1 ∗ · · · ∗
0 ρ n2 · · · ∗
ρ π,p | Gp ∼ ⎜
⎟
⎝ 0 0 · · · ∗ ⎠ ,
0 0 0 ρ nr
65

where each ρ ni is an n i -dimensional representation with shape similar to that in Theorem
5.3.8. In particular, ρ π,p | Gp is quasi-ordinary.
5.6 Further remarks
Theorem 5.5.6 treats the general ordinary case. In the general non-ordinary case, the
behaviour of the (p, p)-Galois representation is more complex and we do not have complete
information about irreducibility (compare with Theorem 5.4.18). In this section we
content ourselves with a few concluding remarks about new issues that arise.
To fix ideas we assume that the Weil-Deligne representation associated to π p is of the
form
χ 1 ⊗ Sp(2) ⊕ χ 2 ⊗ Sp(2),
where χ 1 and χ 2 are unramified characters of W p taking arithmetic Frobenius to α 1 and
α 2 , respectively. Let D be the associated (ϕ, N, Q p , E)-module. Let β 4 > · · · > β 1 be the
jumps in the Hodge filtration on D. We continue to assume that Assumption 4.4.6 holds.
Just as in previous sections, we ignore the filtration, and first classify the (ϕ, N, Q p , E)-
submodules of D.
Classification of (ϕ, N)-submodules of D
Let e 1 , e 2 = N(e 1 ) be a basis of χ 1 ⊗ Sp(2) and f 1 , f 2 = N(f 1 ) be a basis of χ 2 ⊗ Sp(2).
Sometimes we write 〈e 2 〉 for χ 1 ⊗ Sp(1), etc.
If α 1 = α 2 , then any 1-dimensional subspace of 〈e 2 , f 2 〉 is a (ϕ, N)-submodule of
D. In particular, there exists infinitely many 1-dimensional submodules of D. This is
already in striking contrast with the statement of Theorem 5.4.1, which says that in
the indecomposable case there are only finitely many (ϕ, N)-submodules. So already we
can expect that the analysis in the decomposable case might be much more complex.
We remark, however, that if α 1 ≠ α 2 , then 〈e 2 〉 and 〈f 2 〉 are the only 1-dimensional
submodules of D.
Again, if α 1 = α 2 , then the 2-dimensional (ϕ, N)-submodule of D are of the form
〈ae 1 + cf 1 , ae 2 + cf 2 〉, for some a, c ∈ E. If α 1 ≠ α 2 , but p/α 1 = 1/α 2 , then again
there are again infinitely many 2-dimensional (ϕ, N)-submodules and they are given by
〈e 1 + bf 2 , e 2 〉 or 〈f 1 + be 2 , f 2 〉, for some b ∈ E. If α 1 ≠ α 2 and p/α 1 ≠ 1/α 2 , then there
are only finitely many 2-dimensional (ϕ, N)-submodules: they are χ 1 ⊗ Sp(2), χ 2 ⊗ Sp(2)
and the diagonal one χ 1 ⊗ Sp(1) ⊕ χ 2 ⊗ Sp(1).
Finally, like the 1-dimensional case, if α 1 = α 2 , then all the 3-dimensional (ϕ, N)-
submodules of D are of form 〈ae 1 + bf 1 , e 2 , f 2 〉, for any a, b ∈ E. If α 1 ≠ α 2 , there are
exactly two 3-dimensional submodules, namely χ 1 ⊗ Sp(2) ⊕ χ 2 ⊗ Sp(1) and χ 1 ⊗ Sp(1) ⊕
χ 2 ⊗ Sp(2).
Hence, if we choose α 1 and α 2 generically (i.e., α 1 ≠ α 2 and α 1 ≠ pα 2 ), then there are
only finitely many (ϕ, N)-submodules of D, otherwise there are infinitely many (ϕ, N)-
submodules of D. The following table contains the possible Newton numbers of the
(ϕ, N)-submodules D ′ of D.
66

dim E D ′ t N (D ′ ) when α 1 ≠ α 2 t N (D ′ ) when α 1 = α 2
1 −v p (α 1 ), −v p (α 2 ) −v p (α 1 )
2 −v p (α 1 ) − v p (α 2 ), 1 − 2v p (α 1 ), 1 − 2v p (α 2 ) 1 − 2v p (α 1 )
3 1 − 2v p (α 1 ) − v p (α 2 ), 1 − v p (α 1 ) − 2v p (α 2 ) 1 − 3v p (α 1 )
4 1 − 2v p (α 1 ) + 1 − 2v p (α 2 ) 2 − 4v p (α 1 )
An irreducible example
We can now easily construct examples such that the crystal D is irreducible. For instance,
choose any α 1 ∈ E with v p (α 1 ) = 0 and take α 2 = α 1 . Take (β 4 , β 3 , β 2 , β 1 ) = (2, 1, 0, −1).
Using the table above, one can easily check that there are no admissible (ϕ, N)-submodules
of D, except for D itself. We note that since α 1 = α 2 , there are infinitely many (ϕ, N)-
submodules of D, but only finitely many conditions to check for non-admissibility.
All complete flags cannot be reducible
In proving Theorem 5.5.6 we showed that ordinariness implies that a particular complete
flag is admissible. We now wish to point out that not all complete flags in D are necessarily
admissible, even under the ordinariness assumption. Indeed, in the setting of the example
of this section, if we choose α 1 and α 2 such that v p (α 1 ) ≠ v p (α 2 ), then any two complete
flags whose 1-dimensional subspaces are 〈e 2 〉 and 〈f 2 〉, respectively, cannot be admissible
simultaneously, since (under Assumption 4.4.6) we have both β 1 = −v p (α 1 ) and β 1 =
−v p (α 2 ).
Intermediate cases
Finally, in the general decomposable case, the regularity (distinct Hodge-Tate weights) of
the (p, p)-Galois representation ρ π,p | Gp does not imply that it is either (quasi)-ordinary
in the sense of Definition 4.4.7 or irreducible (compare with Theorem 5.4.18). We now
give an example of such an ‘intermediate case’, i.e., an example for which the (p, p)-
Galois representation is reducible, but such that there is no complete flag of reducible
submodules.
Let D be as above. Choose α 1 and α 2 such that v p (α 1 ) = 1 and v p (α 2 ) = −10.
Take (β 4 , β 3 , β 2 , β 1 ) = (17, 4, 0, −1). From the table above, we see that χ 1 ⊗ Sp(1),
χ 1 ⊗Sp(2), and D, are admissible and all the other (ϕ, N)-submodules satisfy the condition
that their Hodge numbers are less than or equal to their Newton numbers. So D is
reducible. However, since α 1 ≠ α 2 and α 1 ≠ pα 2 , there are only two 3-dimensional
(ϕ, N)-submodules of D, and again from the table we see that neither is admissible.
Hence, there is no admissible complete flag of (ϕ, N)-submodules of D.
67

Chapter 6
2-adic Hida theory and
Applications
Hida theory for the prime p is the theory [Hid86b], [Hid86a] that deals with p-ordinary
cuspidal families. While generalizations are now available, for automorphic forms on
groups other than GL 2 and base fields other than Q, many authors tend to shy away from
the prime p = 2. We show that the basic results in the literature stated for odd primes p
remain valid for the prime p = 2. We apply these results to study the local semisimplicity
of ordinary modular Galois representations.
In §6.1, we briefly recall some useful results about group and sheaf cohomology. In
§6.2, we study the relations between various cohomology groups with coefficients. In
§6.3, we state a control theorem for cohomology groups of modular curves and §6.4, §6.5
contain its proof. Since some results that we need from the theory of mod 2 modular
forms are not known, we replace them with other ingredients (cf. Theorem 6.5.3 and
§6.6) to prove the control theorem above for p = 2. Finally, in §6.8, we show that Hida’s
control theorem for the ordinary Λ-adic Hecke algebra holds in this setting.
In §6.9, we deduce a uniqueness theorem for 2-stabilized ordinary cuspidal newforms
in Hida families. Namely, we show that such forms live in unique primitive 2-ordinary
cuspidal families, up to Galois conjugacy, a well-known result when p is odd.
Towards the end (cf. §6.10), we show that a recent pretty application of Hida theory
for odd primes p, namely to understanding the local semisimplicity of p-ordinary modular
Galois representations, continues to hold for the prime p = 2. Following the approach
of [GV04] for odd primes p, and assuming that a modularity result of Buzzard [Buz03] for
Artin-like representations holds in sufficient generality for the prime p = 2, we show that
almost all arithmetic members of a primitive non-CM 2-ordinary cuspidal family have
locally non-split Galois representations. The uniqueness result mentioned above implies
that none of these possibly finitely many exceptions are CM forms.
6.1 Background
In this section, we closely follow the exposition in [Hid86a]. Let GL 2 (R) act on the upper
half plane H as follows. If α ∈ GL + 2 (R), then we let α act on H through linear fractional
transformations. Let ɛ = ( )
1 0
0 −1 act on H by ɛ(z) = −¯z. If A ∈ GL2 (R) with det(A) < 0,
69

then we let A act H using the decomposition A = ɛ(ɛA). Let ι : M 2 (R) → M 2 (R) be
the involution defined by α + α ι = Tr(α), i.e., α ι = Adj(α). Let ∆ be a semi-group in
GL 2 (Q) and ∆ ι denote ι(∆). Let M be Z[∆ ι ]-module and Φ be a congruence subgroup
of SL 2 (Z) contained in ∆ ∩ ∆ ι . Thus M becomes a left Φ-module. The abstract Hecke
ring is denoted by R(Φ, ∆) (cf. [Shi71, Chap. 4], for more details).
6.1.1 Group and sheaf cohomology
A map u of Φ to M is called a 1-cocycle, if u satisfies u(αβ) = u(α) + α · u(β), for all
α, β ∈ Φ. Let Z(Φ, M) denote the group of all 1-cocycles of Φ with values in M, and
B(Φ, M) the subgroup of Z(Φ, M) consisting of maps of the form u(α) = (α − 1)x, for
x ∈ M. Then the usual first cohomology group for M is defined by
H 1 (Φ, M) := Z(Φ, M)/B(Φ, M).
Now we shall define the parabolic cohomology group H 1 p(Φ, M) as follows: Let U be
any unipotent subgroup of SL 2 (Q) and put Φ U = {±1}U ∩Φ. Then we have the restriction
map res U : H 1 (Φ, M) → H 1 (Φ U , M). We shall define
H 1 p(Φ, M) := {c ∈ H 1 (Φ, M) | res U (c) = 0 for all unipotent subgroups U}.
Let U ∞ be the standard unipotent subgroup { ( 1 u
0 1
)
| u ∈ Q}. Every unipotent subgroup
U of SL 2 (Q) can be written as αU ∞ α −1 with α ∈ SL 2 (Z). Then, the correspondence
U ↦→ α(∞) ∈ P 1 (Q) gives a bijection between unipotent subgroups and cusps for SL 2 (Z).
For each cusp s ∈ P 1 (Q), we denote the corresponding unipotent subgroup by U s . We
write Φ s for Φ Us . Another useful description of Φ s , which we use more frequently, is given
by Φ s = {α ∈ Φ | α(s) = s}.
Let C(Φ) be a representative (finite) set for the Φ-equivalence classes of cusps. For
each cusp s ∈ P 1 (Q), there exists γ ∈ Φ such that s 0 = γ(s) ∈ C(Φ), hence γΦ s γ −1 = Φ s0 .
This observation simplifies the definition of the parabolic cohomology group, i.e., there
exists a short exact sequence
0 → H 1 p(Φ, M) → H 1 (Φ, M) → G 1 (Φ, M), (6.1.1)
where G i (Φ, M) = ⊕ s∈C(Φ) H i (Φ s , M), and the last arrow sends each cohomology class to
the sum of its restrictions to Φ s .
To define the sheaf cohomology, we assume that Φ is torsion-free, e.g., Φ = Γ 1 (M)
with M ≥ 4. Write Y for the open complex manifold Φ\H. We give the Φ-module M the
discrete topology and define F (M) = Φ\(H × M). Then F (M) is an étale covering of Y ,
and we can consider the sheaf of continuous sections of F (M) over Y , which we denote by
the same symbol F (M). Then, we consider the usual cohomology group H 1 (Y, F (M)) and
that of compact support H 1 c(Y, F (M)). We shall define the parabolic sheaf cohomology
group H 1 p(Y, F (M)) by the image of H 1 c(Y, F (M)) in H 1 (Y, F (M)).
There are well-known isomorphisms (cf. [Hid81, Prop. 1.1]), which make the following
diagram commutative:
H 1 (Y, F (M))
∼ H 1 (Φ, M)
(6.1.2)
H 1 p(Y, F (M))
∼
H 1 p(Φ, M).
70

6.1.2 Hecke action on the cohomology groups
We shall define the action of the Hecke ring R(Φ, ∆) on the cohomology groups H 1 (Φ, M),
H 1 p(Φ, M), and G 1 (Φ, M). Let Φ ′ be another congruence subgroup contained in ∆ ∩ ∆ ι .
We shall define an operator [ΦαΦ ′ ] : H 1 (Φ, M) → H 1 (Φ ′ , M) for each double coset ΦαΦ ′
in ∆. Decompose the double coset ΦαΦ ′ = ∪ i Φα i . The number of left cosets of Φ in
ΦαΦ ′ is finite and for each γ ∈ Φ ′ , by definition, we can find γ i ∈ Φ such that γ i α j = α i γ
for some α j . Then we can define a map v : Φ ′ → M by v(γ) = ∑ i αι i · u(γ i). Thus the
correspondence [u] ↦→ [v] defines a morphism:
[ΦαΦ ′ ] : H 1 (Φ, M) → H 1 (Φ ′ , M),
such that it preserves parabolic cocyles, and hence [ΦαΦ ′ ] acts on H 1 p(Φ, M). In particular
when Φ ′ = Φ, the Hecke ring R(Φ, ∆) acts on H 1 (Φ, M) and H 1 p(Φ, M). Similarly, one
can define an operator [ΦαΦ ′ ] : H 0 (Φ, M) → H 0 (Φ ′ , M) by x|[ΦαΦ ′ ] = ∑ i αι i · x, for each
x ∈ H 0 (Φ, M). Hence, the Hecke ring R(Φ, ∆) acts on H i (Φ, M) for each i = 0, 1.
Now we introduce a morphism [ΦαΦ ′ ] : G i (Φ, M) → G i (Φ ′ , M) for i = 0, 1. If the
number of left Φ t -cosets in Φ t αΦ ′ s for t ∈ C(Φ) and s ∈ C(Φ ′ ) is finite, we can define a
morphism [Φ t αΦ ′ s] : H i (Φ t , M) → H i (Φ ′ s, M) for i = 0, 1 in the same manner as above. Fix
s ∈ C(Φ ′ ) and write ΦαΦ ′ = ∪ disj Φβ i Φ ′ s, for some β i ’s and write each Φβ i Φ ′ s = ∪ disj Φβ i π j ,
for some π j ∈ Φ ′ s.
Lemma 6.1.1 ([Hid86a], Lem. 4.1). Let t = β i (s). Then the union ∪ j Φ t β i π j coincides
with Φ t β i Φ ′ s and is disjoint. Especially, the number of left cosets in Φ t β i Φ ′ s is finite.
For a given s ∈ C(Φ ′ ), decompose ΦαΦ ′ as above. By definition, we can find γ ∈ Φ
so that γβ i (s) ∈ C(Φ). Thus, we may assume that β i (s) ∈ C(Φ) by substituting γβ i for
β i , if necessary. Then, by the last Lemma, we can define a morphism
[Φ t βΦ ′ s] : H i (Φ t , M) → H i (Φ ′ s, M),
for t = β i (s). For c ∈ G i (Φ, M) (resp., G i (Φ ′ , M)), let us write c t (resp., c s ) for the
component of c in H i (Φ t , M) (resp., H i (Φ ′ s, M)). Then, we shall define
(c|[ΦαΦ ′ ]) s = ∑ i
c βi (s)|[Φ βi (s)β i Φ ′ s].
The operator defined above depends only on the double coset ΦαΦ ′ and via this action
the module G i (Φ, M) becomes an R(Φ, ∆)-module.
We now briefly recall the action of double cosets on sheaf cohomology groups (cf.
[Hid81, §3], for more details). Assume that Φ ′ is torsion-free. For each α ∈ ∆, we put
Φ α = Φ ′ ∩ α −1 Φα, Φ α = αΦ α α −1 ,
and Y ′ = Φ ′ \H, Y α = Φ α \H, Y α = Φ α \H. Then the map α : H × M → H × M defined by
α(z, v) = (α −1 (z), α ι v) induces a morphism [α] : F (M)| Y α → F (M)| Yα , which gives rise
to a morphism [α] : H i (Y α , F (M)) → H i (Y α , F (M)). Since Y α /Y ′ is an étale covering,
we have the trace map Tr Yα/Y ′ : Hi (Y α , F (M)) → H i (Y ′ , F (M)) and the restriction map
Res Y α /Y : H i (Y, F (M)) → H i (Y α , F (M)). We shall define the action of double coset
[ΦαΦ ′ ] : H i (Y, F (M)) → H i (Y ′ , F (M))
by Tr Yα/Y ′ ◦ [α] ◦ Res Y α /Y . In exactly the same manner, we define the action of [ΦαΦ ′ ] on
H i c(Y, F (M)) and H i (Y, F (M)). These actions are compatible with the isomorphism (6.1.2).
71

6.1.3 Geometry of Riemann surfaces
As before, assume that Φ is torsion-free. Take a point y ∈ Y . Then Φ can be identified
with the fundamental group π 1 (Y ) of Y with the base point y. Let X be the smooth
compactification of Y at cusps. Let g denote the genus of X. We choose 2g-curves
{α 1 , β 1 , . . . , α g , β g } passing through y, but not crossing any cusps of X, which form a
system of canonical generators of the group π 1 (X). Namely, π 1 (X) is isomorphic to the
quotient of the free group generated by {α 1 , β 1 , . . . , α g , β g } by the unique relation
[α g , β g ] . . . [α 1 , β 1 ] = 1,
where [α g , β g ] denote the commutator of α g and β g . By cutting X along these 2g-curves,
we have a simply connected polygon of 4g-sides, and inside the polygon, there are cusps
of X. Write X − Y = {x 1 , . . . , x d }, and draw curves π i on the polygon from y encircling
each cusp x i and assume that they intersect only at y. Then, Φ = π 1 (Y ) is generated by
{π 1 , . . . , π d } and {α 1 , β 1 , . . . , α g , β g }, with the only relation π d · · · π 1 · [α g , β g ] · · · [α 1 , β 1 ] =
1. Let Φ ab be the free Z-module generated by {π 1 , . . . , π d , α 1 , β 1 , . . . , α g , β g } and Φ ∞ ab be
the free submodule of Φ ab generated by {π 1 , . . . , π d }. For each Z-module A, let Φ act on
A trivially. Then we have a natural commutative diagram
H 1 (Y, A)
∼ H 1 (Φ, A)
∼
{ϕ ∈ Hom(Φ ab , A) | ∑ d
i=1 ϕ(π i) = 0}
(6.1.3)
H 1 (X, A)
∼
H 1 p(Φ, A)
∼
{ϕ ∈ Hom(Φ ab , A) | ϕ(π i ) = 0 ∀ i}
Put P (Φ) = {π 1 , . . . , π d }. By definition, we can identify P (Φ) with the set of generators
of (the free part of) Φ s for s ∈ C(Φ). Then, we have
H 1 (Φ, A) ≃ H 1 (Φ, Z) ⊗ Z A, H 1 p(Φ, A) ≃ H 1 p(Φ, Z) ⊗ Z A, and,
G(Φ, A) = H 1 (Φ, A)/H 1 p(Φ, A) ≃ {ϕ ∈ Hom(Φ ∞ ab , A) |
6.1.4 Hida’s idempotent operator
∑
π∈P (Φ)
ϕ(π) = 0}.
Let us now introduce the idempotent operator attached to T p in the Hecke algebra. Suppose
R is a commutative sub-algebra of an endomorphism algebra End(M), where M is a
finite free Z p -module. Let T be an element of R. The algebra R/pR is finite-dimensional
over F p . Hence, the image ˜T of T in R/pR can be decomposed into the unique sum s + n
of a semisimple element s and a nilpotent element n. Thus, for a sufficiently large r, the
element ˜T pr coincides with s pr and becomes semisimple. Then, we can choose a positive
integer u so that ˜T pru gives an idempotent of R/pR. This idempotent can be lifted to a
unique idempotent e of R. In fact, this idempotent can be given as a p-adic limit in R by
e = lim
r→∞
T pru .
This idempotent is independent of the choice of the integer u.
Let K be a finite extension of Q p and O K be the integral closure of Z p in K. For r ≥ 1
and k ≥ 2, let h k (Γ 1 (Np r ), O K ) denote the Hecke algebra of level Np r and of weight k.
72

Applying the above discussion with R = h k (Γ 1 (Np r ), O K ), we get an idempotent operator
e r which is compatible with the operator e r−1 . Thus we can define an idempotent e of
lim
←−r≥1 h k(Γ 1 (Np r ), O K ). For modules M over the Hecke algebras, we define the ordinary
part of M to be eM. Sometimes, we denote the ordinary part of M by M 0 .
6.2 Relations between cohomology groups with coefficients
When we study the ordinary parts of the cohomology groups of Γ 1 (Np r ) with (p, N) =1,
for different r’s, we also need to work with the ordinary parts of cohomology groups of
Φ s r, for r ≥ s ≥ 0, where
{( )
}
Φ s r := Γ 1 (Np s ) ∩ Γ 0 (p r a b
) = ∈ SL 2 (Z) | c ≡ 0 (mod Np r ), a ≡ 1 (mod Np s ) ,
c d
{( )
}
∆ s a b
r := ∈ M 2 (Z) | ad − bc > 0, c ≡ 0 (mod Np r ), a ≡ 1 (mod Np s ) .
c d
When studying the action of the Hecke operators on (usual or parabolic) cohomology
groups, often one needs to decompose them into disjoint union of left cosets with a clever
choice of coset representatives. Such useful decompositions can be found in [Hid86a, Lem.
4.3]. We recall with proof only part (ii) of that lemma, since the hypotheses of the original
statement are mildly misstated and the remaining parts are stated as in that lemma.
Lemma 6.2.1. Let r, m ≥ 1, r ≥ s. For every integer u ∈ Z, let α u ∈ M 2 (Z) be such
that
α u ≡ ( 1 u
(mod Np max(m,r) ) and det(α u ) = p m .
0 p m )
Then we have a disjoint decomposition
Φ s ( 1 u
)
r 0 p m Φ
s
r = ⋃
Φ s rα u .
u mod p m
Proof. Suppose that m ≥ r. The proof in the other case is similar. Take Γ ′ in (3.3.2) of
[Shi71, p. 67], as Φ s r, h to be the kernel of (Z/Np r ) × → (Z/Np s ) × and N to be Np r .
By [Shi71, Prop. 3.33], we have that Φ s ( 1 0
)
r 0 p m Φ
s
r = {β ∈ ∆ ′ | det(β) = p m }, where ∆ ′
is as in [Shi71, p. 68]. Since the number of left cosets of Φ s r in Φ s ( 1 u
)
r 0 p m Φ
s
r is p m , and
u ≡ u ′ (mod p m ) if and only if α u ≡ α u ′ (mod p m ), we see that the lemma follows.
Lemma 6.2.2 ( [Hid86a], Lem. 4.3). 1. For each r, s and m satisfying r − s ≥ m > 0
and s ≥ 1, we have: Φ s ( 1 0
)
r 0 p m Φ
s
r = Φ s ( 1 0
)
r 0 p m Φ
s
r−m . In particular, we have
Φ s ( 1 0
)
r 0 p r−s Φ
s
r = Φ s ( 1 0
)
r 0 p r−s Φ
s
s .
2. For each prime l, we have a disjoint decomposition
⎧ (
Φ s 1 0
) ⎨ ∪
r(
0 l Φ
s u mod l Φs 1 u
) ( )
r 0 l ∪ Φ
s
r σ l 0 l 0 1 if l ∤ Np,
r =
)
⎩ ∪
if l | Np,
and
u mod l Φs r
( 1 u
0 l
Φ s rlσ l Φ s r = Φ s rlσ l if l ∤ Np,
where σ l is an element of SL 2 (Z) satisfying σ l ≡ ( l −1 ∗
0 l
)
(mod Np r ).
73

3. Take δ ∈ SL 2 (Z) such that δ ≡ ( )
0 1
−1 0 (mod p 2r ) and δ ≡ 1 (mod N 2 ). We have a
disjoint decomposition:
Φ 0 rδΦ 0 r = ∪ rδ ( )
1 u
u mod p rΦ0 0 1 .
We now introduce certain important modules and we later study the relations between
cohomology groups with these as coefficients. Firstly, we consider the column vector space
L n (Z) = Z n+1 for each non-negative integer n. Let (x, y) t be a variable vector in L 1 (Z)
and define
( ) x n
= (x n , x n−1 y, . . . , y n ) ∈ L n (Z).
y
We let M 2 (Z) act on L n (Z) through the symmetric n-th tensor representation explicitly
given by ( ) ( )
a b x n ( ) ax + by n
· =
.
c d y cx + dy
For any Z-module A, put L n (A) = L n (Z) ⊗ Z A with the natural action of M 2 (Z) on the
left factor. For Φ = Γ 1 (Np r ), we take ∆ = M 2 (Z) ∩ GL 2 (Q) (cf. §6.1). Let K be a finite
extension of Q p and let O denote the integral closure of Z p in K. For a natural number
t, we define the map χ t from (Z/p r Z) × → (Z/p r Z) × ⊂ (O/p r O) × by χ t (a) = a t . Finally,
let χ be a character from (Z p /p r Z p ) × to either O × or (O/p t O) × .
We shall define a twisted action of α = ( )
a b
c d ∈ (∆
s
r ) ι on L n (O/p t O) by α · x =
χ(d)(αx) for x ∈ L n (O/p t O), where αx is the usual action. We denote this twisted
module by L n (χ, O/p t O). In the theorem below, we relate the cohomology groups for the
module L n (O/p r O) with those of the module L 0 (χ n , O/p r O). For every natural number
r, we define
i r : L n (O/p r O) → L 0 (χ n , O/p r O)
and
(x 0 , . . . , x n ) → x n
j r : L 0 (χ −n , O/p r O) → L n (O/p r O)
x 0 → (x 0 , 0, . . . , 0)
These are Γ 0 (p r )-module homomorphisms and they covariantly induces the maps:
ι r := (i r ) ∗ : H 1 (Φ s r, L n (O/p r O)) → H 1 (Φ s r, L 0 (χ n , O/p r O))
and
(j r ) ∗ : H 1 (Φ s r, L 0 (χ −n , O/p r O)) → H 1 (Φ s r, L n (O/p r O)).
Let τ ∈ M 2 (Z) such that det(τ) = p r and,
τ ≡ ( )
0 −1
p r 0
(mod p 2r ) and τ ≡ ( 1 0
0 p r )
(mod N 2 ).
One checks that [τ] induces an isomorphism between the groups H 1 (Φ s r, L 0 (χ, O/p r O))
and H 1 (Φ s r, L 0 (χ −1 , O/p r O)) defined as follows: For each cocycle u, let (u|[τ])(γ) =
θ(u(τγτ −1 )), where θ is an isomorphism between L 0 (χ, O/p r O) ≃ L 0 (χ −1 , O/p r O). Let
δ ∈ SL 2 (Z) be such that
δ ≡ ( )
0 1
−1 0
(mod p 2r ) and δ ≡ ( )
1 0
0 1
74
(mod N 2 ).

We define a new morphism H 1 (Φ s r, L 0 (χ n , O/p r O)) → πr
H 1 (Φ s r, L n (O/p r O)), which is the
composition of the following maps
[τ]
H 1 (Φ s r, L 0 (χ n , O/p r O)) H 1 (Φ s r, L 0 (χ −n , O/p r O))
(j r) ∗
H 1 (Φ s r, L n (O/p r O)) H 1 (Φ s r, L n (O/p r O)).
[Φ s rδΦ s r]
The abstract Hecke ring R(Φ s r, ∆ s r) acts on the cohomology groups for L n (A) as follows.
For each prime l, the Hecke operators T (l) and T (l, l) act on the cohomology groups for
L n (A) by the action of double cosets (cf. §6.1.2), i.e., T (l) := [ Φ s 1 0
) ]
r(
0 l Φ
s
r and
{
[Φ
s
r lσ l Φ s r] if l ∤ Np r ,
T (l, l) :=
0 if l | Np r ,
where σ l is an element of SL 2 (Z) such that σ l ≡ ( l −1 ∗
0 l
)
(mod Np r ).
Theorem 6.2.3. For each positive integer r, with r > s ≥ 1, we have:
π r ◦ ι r = T (1, p r ) on H 1 (Φ s r, L n (χ, O/p r O)),
and
ι r ◦ π r = T (1, p r ) on H 1 (Φ s r, L 0 (χ n , O/p r O)).
Moreover, the maps ι r , π r preserve parabolic classes and the map ι r is T p -equivariant.
Proof. We show that ι r is equivariant under the action of T = T (l) or T (l, l), for all l.
By Lemma 6.2.2 (2), we can decompose T as disjoint union of left cosets:
T = ∪ i Φ s rα i with α i ≡ ( )
1 ∗
0 ∗
(mod p r ).
If l | Np, then T (l, l) is zero. If l ∤ Np, then T (l, l) = [Φ s rlσ l Φ s r] = [Φ s rlσ l ] = Φ s rlσ l .
Now, by the choice of σ l , we see the above decomposition. A similar argument holds for
T (l) also. Then, on L n (O/p r O), αi ι acts by the matrix of the form:
⎛
⎞
∗ ∗ · · · ∗
⎜
⎟
⎝· · · · · · · · · ∗⎠ ∈ M n+1 (O/p r O).
0 0 · · · 1
To check the map ι r is T p -equivariant, we need to show that ι r ◦ (u|T ) = (ι r ◦ u)|T . For
each 1-cocycle u : Φ s r → L n (O/p r O), we have
ι r ((u|T )(γ)) = ι r ( ∑ i
α ι i.u(γ i )) = ∑ i
ι r (u(γ i )) = ∑ i
(ι r ◦ u)(γ i ) = ((ι r ◦ u)|T )(γ),
where γ i ∈ Φ s r is defined by the relation α i γ = γ i α j for some j.
We now prove the identity π r ◦ ι r = T (1, p r ). By definition, we have:
(π r ◦ ι r (u))(γ) = ([Φ s rδΦ s r] ◦ (j r ) ∗ ◦ [τ] ◦ ι r (u))(γ) =
p∑
r −1
a=0
δ ι a · j r (ι r (τγ a τ −1 )),
(∗)
75

where δ a = δ ( )
1 a
0 1 for δ ∈ SL2 (Z) such that δ ≡ ( )
0 1
−1 0 (mod p 2r ) and δ ≡ 1 (mod N 2 ).
The element γ a is defined by the equation δ a γ = γ a δ b for some b with 0 ≤ b < p r . Since
τ ≡ ( )
0 −1
0 0 (mod p r ), we have τ ι · x = (x n , . . . , 0) t , for x = (x 0 , . . . , x n ) ∈ L n (O/p r O),
and therefore τ ι · x = j r (i r (x)). The expression in (∗) reduces to
(π r ◦ ι r (u))(γ) =
p r −1
∑
(τδ a ) ι · j r (ι r (τγ a τ −1 )).
a=0
We have τδ a ≡ ( )
1 a
0 p r (mod Np r ), det(α a ) = p r , and (τδ a )γ = (τγ a τ −1 )(τδ b ). By
Lemma 6.2.1, we see that the required identity holds.
We now check the identity ι r ◦ π r = T (1, p r ) on H 1 (Φ s r, L 0 (χ n , O/p r O)). For each
1-cocycle u : Φ s r → L 0 (χ n , O/p r O), we have
p∑
r −1
((ι r ◦ π r )(u))(γ) = ι r ( δa ι · j r (τγ a τ −1 )) =
a=0
p∑
r −1
a=0
ι r (δ ι a · j r (u(τγ a τ −1 ))).
Assume that ι r (δ ι a · j r (x)) = x. Then the expression in (∗∗) implies the identity, because
((ι r ◦ π r )(u))(γ) =
p∑
r −1
a=0
ι r (δ ι a · j r (u(τγ a τ −1 ))) =
p∑
r −1
a=0
u(τγ a τ −1 ) = (u|T (1, p r ))(γ),
by the decomposition of T (1, p r ). The identity ι r (δa ι · j r (x)) = x follows from the fact that
the matrix δa ι ≡ ( )
−a −1
1 0 (mod p r ) acts on L n (O/p r O) by a matrix of the form:
⎛
⎞
∗ ∗ · · · ∗
⎜
⎟
⎝· · · · · · · · · ∗⎠ ∈ M n+1 (O/p r O).
1 0 · · · 0
(∗∗)
6.3 Main theorems
We start this section by introducing some notations, and then we state one of the main
theorems of this chapter. From now on p = 2, q = 4 and (2, N) = 1, unless explicitly
stated. Let Γ 0 = Γ 1 = Z × p and for r ≥ 2, let Γ r denote the subgroup 1 + p r Z p of Γ, where
Γ = Γ 2 = 1 + qZ p = 〈u〉. For r ≥ s ≥ 0, there is a short exact sequence of groups
0 → Γ 1 (Np r ) → Φ s r → Γ s /Γ r → 0, (6.3.1)
induced by Φ s r ∋ ( a b
c d
)
↦→ ¯d ∈ Γs /Γ r . We write Φ r for Φ 0 r, for r ≥ 0. In Hida theory for
odd primes p, the congruence group Φ 1 plays an important role, but when p = 2, the role
of this group is played by the group Φ 2 = Γ 0 (4) ∩ Γ 1 (N), which is torsion-free, if N > 1.
Let K be a finite extension of Q p and O K be the integral closure of Z p in K. By
definition, there is a tautological character ι : Γ ↩→ Λ K = O K [[Γ]], which takes u to itself
in Λ K . For each character χ : Γ → O × K , the element P χ = ι(u) − χ(u) is a prime element,
and the quotient Λ K /P χ Λ K is isomorphic to O K . If χ(u) = ɛ(u)u k , where ɛ is a finite
order character of Γ, we write P k,ɛ for P χ , and simply write P k , if ɛ is trivial. We may
identify Λ K with O K [[X]] sending u to 1 + X. When K = Q p , we denote Λ Qp by Λ.
76

Finally, let ω denote the mod 4 cyclotomic character, that is, ω is the mod 4 character
defined by ω(x) = ±1, for x ≡ ±1 (mod 4).
Recall that p = 2, and N is odd. We only use the congruence subgroup Γ 1 (Np r ) with
r ≥ 2, which is a torsion-free group. We denote the corresponding complex Riemann
surface by Y r and its compactification by X r . Let H i (Y r , M) and H i (X r , M) denote the
corresponding sheaf cohomology groups for each constant sheaf M of Z-modules. It is
well-known that
S 2 (Γ 1 (Np r )) ≃ H 1 (X r , R),
where the right hand side of the isomorphism, the sheaf cohomology with R coefficients,
can be identified with de Rham cohomology. The above isomorphism is invariant under
the Hecke action. The Hecke algebra h 2 (Γ 1 (Np r ), Z) acts on H 1 (X r , Z) and therefore
h 2 (Γ 1 (Np r ), Z p ) acts on H 1 (X r , Z p ), H 1 (X r , Q p ), H 1 (X r , T p ), where H 1 (X r , M) =
H 1 (X r , Z) ⊗ Z M and T p := Q p /Z p .
For every positive integer r ≥ 2, we simply write
V r = H 1 (X r , T p ), W r = H 1 (Y r , T p ).
Since H 1 (X r , T p ) ≃ H 1 (X r , Q p )/H 1 (X r , Z p ), we see that V r and W r are p-divisible modules
of finite Z p -corank. Therefore End(V r ) and End(W r ) are free of finite rank. By §6.1.4, we
can define Hida’s idempotent operator e r attached to the Hecke operator T p in End(V r )
and End(W r ). Define the ordinary parts of V r to be V 0 r = e r V r and similarly for W r .
Therefore, V 0 r is a module for h 0 2 (Γ 1(Np r ), Z p ), the ordinary part of h 2 (Γ 1 (Np r ), Z p ).
There is also an action of Γ 0 (Np r )/Γ 1 (Np r ) on V 0 r and W 0 r . Let V denote the direct
limit of V r and define similarly W, V 0 and W 0 . Since (Z/Np r Z) × acts on V 0 r and W 0 r ,
hence the inverse limit, over r ≥ 2, of (Z/Np r Z) × acts on V 0 and W 0 . In particular V 0
and W 0 become continuous modules over the Iwasawa algebra Λ = Z p [[Γ]], if we equip
them with the discrete topology. Let V 0 and W 0 denote the Pontryagin dual modules of
V 0 and W 0 , respectively. Then V 0 and W 0 are compact Λ-modules. We can now state
one of the main theorems of this chapter.
Theorem 6.3.1. Let p = 2. We have:
1. For each positive integer r ≥ 2, the restriction morphism of cohomology groups
induces an isomorphism of V 0 r onto (V 0 ) Γr . The same result also holds for W 0 .
2. Let N > 1. The modules V 0 and W 0 are free modules of finite rank over Λ.
The first part of Theorem 6.3.1 gives control of the ordinary parts of the cohomology
modules associated with the decreasing sequence of congruence subgroups Γ 1 (N2 r ), for
r ≥ 2, and we refer to such a result as a control theorem (for cohomology).
6.4 Control theorem for cohomology
Before we start the proof of part (1) of Theorem 6.3.1, we shall state a lemma.
Lemma 6.4.1. Let {M r } r≥2 be an inductive system of compatible modules over Z p [Γ/Γ r ],
respectively. Assume that for all r ≥ t ≥ 2, Mr
Γt = M t . Then (lim M −→r r ) Γt = M t .
77

Proof. Clearly, lim M −→r r is a Z p [[Γ]]-module. Since H q (Γ t , lim M −→r r ) = lim H q (Γ −→r t , M r ), for
q ≥ 0, we have (lim M −→r r ) Γt = M t , where the last equality follows from the assumption.
Now, we start the proof of the control theorem for cohomology.
Lemma 6.4.2. If Φ s r/Γ 1 (Np r ) acts on T p trivially, then H 2 (Φ s r/Γ 1 (Np r ), T p ) = 0.
Proof. Since Φ s r/Γ 1 (Np r ) is a finite cyclic group, H 2 (Φ s r/Γ 1 (Np r ), T p ) = T p /N T p , where
N denote the norm map from T p to itself. Since Φ s r/Γ 1 (Np r ) acts on T p trivially, N is
multiplication by the index of Γ 1 (Np r ) in Φ s r, hence is surjective.
Lemma 6.4.3. For each r ≥ s ≥ 2, eH 1 (Γ 1 (Np s ), T p ) ≃ eH 1 (Φ s r, T p ).
Proof. Since Φ s r ⊆ Γ 1 (Np s ), there is a restriction map H 1 (Γ 1 (Np s ), T p ) → H 1 (Φ s r, T p ).
We have the following commutative diagram
H 1 (Γ 1 (Np s ), T p )
H 1 (Φ s r, T p )
Tp
r−s
Tp
r−s
[Φr( s 1 0
0 p
)Φ r−s s s]
H 1 (Γ 1 (Np s ), T p ) H 1 (Φ s r, T p ).
By applying the idempotent operator, we get that the vertical morphisms are isomorphisms
and hence the diagonal map is an isomorphism.
res
For each r ≥ s ≥ 2, we have the inflation-restriction sequence
0 → H 1 (Φ s r/Γ 1 (Np r ), T p )
res
ι
→ H 1 (Φ s r, T p )
→ H 1 (Γ 1 (Np r ), T p ) Γs → H 2 (Φ s r/Γ 1 (Np r ), T p ) = 0
where the last term vanishes by Lemma 6.4.2. The image of the group H 1 (Φ s r/Γ 1 (Np r ), T p )
inside H 1 (Φ s r, T p ) is annihilated by the idempotent e attached to T p by [Hid86a, Lem. 6.1].
Therefore,
eH 1 (Φ s r, T p ) ≃ eH 1 (Γ 1 (Np r ), T p ) Γs = (W 0 r ) Γs .
By Lemma 6.4.3, we have that Ws
0
these isomorphisms, we get
= eH 1 (Γ 1 (Np s ), T p ) ≃ eH 1 (Φ s r, T p ). By combining
W 0 s ≃ (W 0 r ) Γs .
By Lemma 6.4.1, for any r ≥ 2, we have that (W 0 ) Γr ≃ Wr 0 . This finishes the proof of
the control theorem for W 0 . Note that so far the proof works for all primes p ≥ 2.
Now, we prove the control theorem for V 0 , concentrating on what changes need to be
made in Hida’s original proof when p = 2. By (6.1.3), for r ≥ 2, the module W r is given
by {ϕ ∈ Hom(Γ 1 (Np r ), T p ) | ∑ π∈P (Γ 1 (Np r )) ϕ(π) = 0}, and V r is the submodule of W r
given by
{
}
V r = ϕ ∈ Hom(Γ 1 (Np r ), T p ) | ϕ(π) = 0 for π ∈ P (Γ 1 (Np r )) ,
since the group Γ 1 (Np r ) acts trivially on T p .
By Lemma 6.4.1, it is enough to prove that Vs 0 = (Vr 0 ) Γs , for r ≥ s ≥ 2. Since
Ws 0 ≃ (Wr 0 ) Γs , we have Vs 0 ↩→ (Vr 0 ) Γs . Therefore, it is enough to prove the surjectivity of
78

this map. Since (W 0 r ) Γs = W 0 s , given a homomorphism ϕ : Γ 1 (Np r ) → T p invariant under
Γ s and satisfying ϕ| e = ϕ, there exists a homomorphism ψ : Γ 1 (Np s ) → T p satisfying
ψ| e = ψ such that ψ = ϕ on Γ 1 (Np r ). Thus, we need to show that ψ(π) = 0, for all π ∈
P (Γ 1 (Np s )), assuming the same holds for ϕ with r instead of s. Let [ψ] denote the equivalence
class of ψ in the module eG(Γ 1 (Np s ), T p ) := eH 1 (Γ 1 (Np s ), T p )/eH 1 p(Γ 1 (Np s ), T p ).
We need to show that [ψ] = 0. We know that [ψ]| e = [ψ]. If [ψ]| 1−e = [ψ], then [ψ] = 0,
since e is an idempotent. Hence, it is enough to show that
[ψ]| 1−e = [ψ]
holds. By following the strategy in [Hid86a], this reduces to proving [Hid86a, Thm. 5.8],
which characterizes the elements of (1 − e)G(Γ 1 (Np s ), T p ) as elements of the set
V (T p ) := {ψ ∈ Hom(Γ 1 (Np s ) ∞ ab , T p) | ψ(π) = 0, for all π ∈ P (Γ 1 (Np s ))
corresponding to the unramified cusps},
where the module Γ 1 (Np s ) ∞ ab as in §6.1.3. Under the above equality, if ψ ∈ V (T p), then
[ψ] = (1 − e)[ψ ′ ] and hence [ψ]| 1−e = [ψ] holds. Thus it suffices to show that ψ ∈ V (T p ).
Since every unramified cusp of X s over X 0 is under an unramified cusp of X r over X 0 ,
the elements of P (Γ 1 (Np s )) corresponding to unramified cusps in X s can be taken to
be among the elements of P (Γ 1 (Np r )) corresponding to unramified cusps in X r . Then
ψ(π) = ϕ(π) = 0, for all π ∈ P (Γ 1 (Np s )), which corresponds to unramified cusps of
Γ 1 (Np s ), as desired.
The proof of [Hid86a, Thm. 5.8] depends, firstly, on various relations between the
dimensions of the space of Eisenstein series for Γ 1 (Np r ) with coefficients in A and the
boundary cohomology G 1 (Γ 1 (Np r ), L n (A)) (cf. §6.1.1), for any subalgebra A of C or C p ,
and secondly, on the validity of [Hid86a, Prop. 5.7]. The results on the dimensions of
the space of Eisenstein series and the space G 1 (Γ 1 (Np r ), L n (A)) also hold for the prime
p = 2. But, in the proof of [Hid86a, Prop. 5.7], one crucially uses the fact that p ≠ 2.
Thus to finish the proof of the control theorem for V 0 when p = 2, it suffices to check
that the proposition holds.
The difference between [Hid86a, Prop. 5.7] and the analogous result for p = 2 (Proposition
6.4.5 below) is that in the former case, i.e., when Φ = Γ 1 (Np r ) for p ≥ 5 and
r ≥ 1, the group Φ is torsion-free with regular cusps, whereas in the latter case, i.e., when
Φ = Γ 1 (Np r ), for p = 2 and r ≥ 2, the group Φ is torsion-free, but its cusps are not
necessarily regular, for example, when N = 1 and r = 2. When p = 2, these irregular
cusps create problems in the proof given in [Hid86a, Prop. 5.7].
Definition 6.4.4. Let Φ be a congruence subgroup of Γ 1 (Np r ) for r ≥ 2. Let s ∈ C(Φ)
be a cusp and s 0 be the image of s in C(Γ 1 (N)). We say that s is unramified, if s is
unramified over s 0 , in the sense of finite covers of Riemann surfaces.
Proposition 6.4.5. Let Φ = Γ 1 (Np r ) with r ≥ 2. Let A be either Z p , Z p /p i Z p or any field
of characteristic 0. Let s be any unramified cusp of Φ and ρ s : G 1 (Φ, A) → H 1 (Φ s , A)
be the natural projection map. Then for any c ∈ G 1 (Φ, A), we have that ρ s (c|e) = ρ s (c).
Proof. For any positive integer M ≥ 3 with M ≠ 4, all the cusps of Γ 1 (M) are regular.
Hence, when p = 2, all cusps of Γ 1 (Np r ), for N ≥ 3 or r ≥ 3 are regular (so in particular
are the unramified ones). So, it is enough to consider the case when Φ = Γ 1 (4).
79

By [Hid86a, Lem. 5.1], Γ 1 (4) has an unique unramified cusp, namely ∞. We see that this
cusp is also regular since otherwise we would have
π s = π ∞ = − ( 1 u
0 1
)
,
for some u > 0, which is not an element of Γ 1 (4). Hence, when N = 1 and r = 2, the
unique unramified cusp is also regular.
For any cusp s ∈ C(Φ), the group Φ s is an infinite cyclic group and generated by π s .
The evaluation of 1-cocyles at πs
N gives an isomorphism between H 1 (Φ s , A) and A. Thus
H 1 (Φ s , A) ≃ H 1 (Φ s , Z p ) ⊗ Zp A.
Thus, we may assume that A = Z p . Take an integer m ≥ r such that p m ≡ 1 (mod N).
By replacing s by another cusp in the Φ-equivalence of s if necessary, we may suppose
that α(∞) = s with α ∈ Γ 0 (p m ), by [Hid86a, Lem. 5.1]. Then α −1 π s α = ( )
1 u
0 1 with
positive u prime to p, since all the unramified cusps of Φ are regular. We shall choose a
disjoint decomposition:
Φ ( 1 0
0 p m )
Φ = ∪
p m −1
j=0 Φβ j and Φ s β 0 Φ s = ∪ pm −1
j=0 Φ sβ j
with β j ∈ Φ s , for all j, β j π N = β j+1 , if 0 ≤ j < p m − 1, and β p m −1π N = π N β 0 .
Take β j ′ = ( 1 jNu
)
0 p , for 0 ≤ j < p m m . Then, β j ′ ≡ ( )
1 0
0 1 (mod N). If we put βj =
αβ j ′ α−1 , then β j ≡ 1 (mod N). Since α = ( )
a b
c d ∈ Γ0 (p m ), we have β j ≡ ( )
1 −ab+a 2 Nuj
0 0
(mod p m ) and det(β j ) = p m . Since a, N and u are relatively prime to p, we see that
−ab + a 2 Nuj runs over all residues modulo p m , as j varies over all residues modulo p m .
By Lemma 6.2.1 and using the fact that m ≥ r, we have the equality
Φ ( 1 0
0 p m )
Φ = ∪
p m −1
j=0 Φβ j.
Since β ′ j (∞) = ∞, we see that β j(s) = s, for all j. If 0 ≤ j < p m − 1, then
(α −1 β j α)(α −1 πα) N = β j( ′ 1 u
) N (
0 1 = 1 (j+1)Nu
)
0 p = β
′ m j+1 ,
and therefore β j π N = β j+1 . When j = p m − 1, a similar calculation shows that β j π N =
π N β 0 holds. Using the properties of β j ’s, we see that Φβ 0 Φ s = Φ ( )
1 0
0 p m Φ = ∪
p m −1
j=0 Φβ j.
Now, by Lemma 6.1.1, we have the disjoint decomposition:
Φ s β 0 Φ s = ∪ pm −1
j=0 Φ sβ j . (6.4.1)
From the above decomposition, we see that, for each 1-cocycle ϕ : Φ s → A, we have
ϕ|[Φ s β 0 Φ s ](π N ) = ϕ(π N ), which shows that ϕ|[Φ s β 0 Φ s ] = ϕ. Since the Hecke operator
T (p m ) acts on G 1 (Φ, A) by ρ s (c|T (p m )) = ρ s (c)|[Φ s β 0 Φ s ] (cf. §6.1.2), we see that
ρ s (c|T (p m )) = ρ s (c), and hence the proposition.
This finishes the proof of the control theorem for cohomology. We remark that subsequently
we will only need to consider N ≥ 3, for applications to p-adic families.
Before proving the part (2) of Theorem 6.3.1, we prove an important result. The
following proposition was proved in [Hid88a, Prop. 2.3] for Γ 1 (Np r ) when p ≥ 5. We
shall prove a similar result for Φ s r for the prime p = 2 and with N ≥ 3.
80

Proposition 6.4.6. For r ≥ 2, the quotient module eH 1 (Φ s r, L n (Z p ))/eH 1 p(Φ s r, L n (Z p )) is
Z p -free.
Proof. We write ∆ for Φ s r. If eG 1 (∆, L n (Z p )) is Z p -free, then the proposition follows,
since the maps in (6.1.1) are invariant under the action of the Hecke operators. Consider
the long exact sequence of cohomology groups
G 0 (∆, L n (Q p )) β → G 0 (∆, L n (T p )) → G 1 (∆, L n (Z p )) → G 1 (∆, L n (Q p )),
induced by the short exact sequence 0 → L n (Z p ) → L n (Q p ) → L n (T p ) → 0.
Since (2, N) = 1 and N ≥ 3, all the cusps of ∆ are regular, because irregularity for
∆ implies the irregularity for Γ 1 (N), but there are no irregular cusps for Γ 1 (N). Let
s ∈ C(∆) be a cusp of ∆. Let α s = ( )
a b
c d ∈ SL2 (Z) such that α s (∞) = s. If π s denotes a
generator for ∆ s , we may write
(
π s = α 1 u
)
s 0 1 α
−1
s = ( )
1−cau a 2 u
−c 2 u 1+cau ∈ ∆, with u ≠ 0. (6.4.2)
We shall show that the idempotent e annihilates G 0 (∆, L n (T p ))/β(G 0 (∆, L n (Q p ))). Since
the image of β is p-divisible, it is sufficient to know that if x ∈ G 0 (∆, L n (T p ))[p], then
x|T (p) m ∈ Im(β) for sufficiently large m. We shall divide our argument into the following
three cases: case 1: p ∤ u; case 2: p | u and p | c, and case 3: p | u but p ∤ c. Since
c 2 u ≡ 0 mod Np r , c is automatically divisible by p in case 1. We have the following
decomposition:
∆ ( {
)
1 0 ∐
p−1
i=0 ∆( 1 0
0 p
0 p ∆ =
)
π
i
s = ∆ ( 1 0
0 p
)
∆s , in case 1,
∐ p−1
i=0 ∆( 1 i
0 p
)
∆s , in case 2 and 3.
(6.4.3)
The decompositions follow from the proof of the Proposition 6.4.5, since p | c and hence
α s ∈ Γ 0 (p). For an element x ∈ G 0 (∆, L n (T p )), we let x s denote the component of x in
H i (∆ s , L n (T p )).
For an element x ∈ G 0 (∆, L n (T p ))[p], we would like to write down the component
(x|T (p)) s . In order to write down this, first we choose γ ∈ ∆ in case 1 and γ i ∈ ∆ in
)
(s) ∈ C(∆) and γi
( 1 i
0 p
)
(s) ∈ C(∆) and we denote them by
cases 2 and 3, so that γ ( 1 0
0 p
ν i (s) and t respectively. From the definition of action of T (p) on x, we see that
⎧
∑p−1
(γ ( )
1 0
⎪⎨
0 p π
i
s ) ι · x t = (γ ( )
1 0
0 p )ι · x t , in case 1,
i=0
(x|T (p)) s =
∑p−1
(
⎪⎩ (γ 1 i
)
i 0 p )ι · x γi (s), in case 2 and 3.
i=0
(6.4.4)
In case 1, (x|T (p)) s acts on p-torsion points by the nilpotent matrix
⎛
⎞
0 0 · · · ∗
0 0 · · · ∗
⎜
⎟
⎝· · · · · · · · · ∗⎠ , (6.4.5)
0 0 · · · 0
and hence (x|T (p) 2 ) s is zero.
81

In case 2, since a is prime to p, for any i 0 , i 1 , . . . , i m ∈ Z, we see that
( 1 im
0 p
)( 1 im−1
0 p
) (
· · ·
1 i0
) a + c ∑ m
j=0 pj i j
0 p (s) =
p m c
∈ Q.
Define ν ij := ( 1 i j
)
0 p . The numerator of the above number is prime to p, and thus if m ≥ r,
the cusp ν = ν im ◦ ν im−1 ◦ · · · ◦ ν i0 (s) as a cusp of Γ 1 (Np r ) is unramified over X 1 (N) and
hence as a cusp of ∆. Then we have
1. L n (p −1 Z/Z) ∆ν ⊆ {(y, 0) t | y ∈ (p −1 Z/Z) n }. Let α ν = ( )
a b
c d ∈ SL2 (Z) such that
α ν (∞) = ν. Since ν is an unramified regular cusp, we can choose α ν ∈ Γ 0 (p m ) such
that αν
−1 π ν α ν = ( )
1 u
0 1 with (u, p) = 1. If x = (x0 , . . . , x n ) ∈ L n (p −1 Z/Z) ∆ν , then
αν
−1 x is fixed by ( )
1 u
0 1 , which implies that (n + 1)-th co-ordinate of the tuple α
−1
ν x
is zero. Therefore, x n = 0 (here use the fact that α ν ∈ Γ 0 (p m )).
(
2. (γ 1 i
)
i 0 p ) ι x ν = 0, if x ν ∈ L n (p −1 Z/Z) ∆ν = H 0 (∆ ν , L n (T p ))[p]. If γ i = ( )
a b
c d , then
( ) ι
(
(γ 1 i
)
i 0 p ) ι a ia + pb
=
acts on a point as above by the matrix
c ci + pd
⎛
⎞
0 0 · · · (ia) n
0 0 · · · ∗
⎜
⎟
⎝· · · · · · · · · ∗ ⎠ . (6.4.6)
0 0 · · · ∗
Therefore, the element (x|T (p) r+1 ) s = 0. In the last case, the proof is similar to the proof
of [Hid88a, Prop. 2.3].
6.5 Freeness
In this section, we prove that the modules V 0 and W 0 are free of finite rank over Z p [[X]],
completing the proof of Theorem 6.3.1. We restate this formally as:
Theorem 6.5.1. Let p = 2, N > 1 be odd, and Λ = Z p [[X]]. The modules V 0 and W 0
are free modules of finite rank over Λ.
A proof of the freeness of W 0 is proved in Appendix A (cf. Theorem A.5.3). The claim
for V 0 is more subtle and requires more machinery and results. We start by recalling
without proof a lemma [Hid86a, Lem. 6.3] which is useful in proving the freeness of V 0 .
Lemma 6.5.2. A compact continuous Λ-module M is free of finite rank r over Λ if and
only if there is a subset I of positive integers and infinitely many elements {P n } n∈I in
Λ such that M[P n ] ≃ T r p, for all n ∈ I, where M is the Pontryagin dual of M and
M[P n ] = {m ∈ M|P n .m = 0}.
We know that the group Z × p (recall p = 2) acts on V 0 . In particular, µ 2 = (Z/qZ) ×
acts on V 0 (recall q = 4). Write
V 0 = V 0 (0) ⊕ V 0 (1),
82

where V 0 (a) = {v ∈ V 0 | v|ζ = ζ a v, for ζ ∈ µ 2 }. Since the action of Γ commutes with the
action of µ 2 , V 0 (a) is also a Λ-module, for a = 0, 1. Let V 0 (a) denote the Pontryagin
dual of V 0 (a). We shall show V 0 (a) is a free module of rank 2r(a) over Λ, where r(a) is
the rank of the Hecke algebra h 0 2 (Φ 2, ω a , Z p ).
By part (1) of Theorem 6.3.1, we have that V 0 (a)/a 2 V 0 (a) ≃ V2 0(a),
where a 2 is the
augmentation ideal of Z p [[Γ]]. Since V2 0(a)
is a free module of rank 2r(a) over Z p, by
Nakayama’s lemma, we see that V 0 (a) is a finitely generated Λ-module with minimal
number of generators 2r(a). Hence there is a surjection from Λ 2r(a) ↠ V 0 (a). Hence, by
duality, we have
V 0 (a)[P n ] ↩→ T 2r(a)
p ,
where P n is the prime ideal of Λ defined in §6.3. Now define
H 1 (Γ 1 (Np r ), n; Z p /p r Z p ) := {v ∈ H 1 (Γ 1 (Np r ), Z p /p r Z p ) | v|z = z n v for z ∈ Z × p },
H 1 p(Γ 1 (Np r ), n; Z p /p r Z p ) := H 1 (Γ 1 (Np r ), n; Z p /p r Z p ) ∩ H 1 p(Γ 1 (Np r ), Z p /p r Z p ).
Suppose that the following inclusions and isomorphisms are true for r ≥ 2:
eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /p r Z p ↩→
(1)
eH 1 p(Φ 2 , L n (Z p /p r Z p ))
≃
(2) eH1 p(Φ r , L n (Z p /p r Z p )) ≃ eH 1
(3)
p(Φ r , Z p /p r Z p (n))
↩→
(4)
eH 1 p(Γ 1 (Np r ), n; Z p /p r Z p ) ↩→
(5)
V 0 (a)[P n ],
(6.5.1)
where the last inclusion holds only if n ≡ a (mod 2). Then we have
eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /p r Z p ↩→ V 0 (a)[P n ] ↩→ T 2r(a)
p .
Taking direct limits with respect to r, we have that
eH 1 p(Φ 2 , L n (Z p )) ⊗ T p ↩→ V 0 (a)[P n ] ↩→ T 2r(a)
p . (6.5.2)
In the next section, we prove that eH 1 p(Φ 2 , L n (Z p )) is Z p -free (cf. Lemma 6.6.2). More
precisely, we prove in Theorem 6.6.1 that:
Theorem 6.5.3. The Z p -rank of the module eH 1 p(Φ 2 , L n (Z p )) is 2r(a), for n ≡ a (mod 2).
Proof. For p ≥ 5, the theorem follows from [Hid86b, Thm. 3.1 and Cor. 3.2], but
their proofs use results from the theory of Katz modular forms and the theory of mod p
modular forms. For the prime p = 2, we need different arguments to prove the theorem
(cf. Theorem 6.6.1) and we postpone the proof to the next section.
We complete the proof of Theorem 6.5.1, assuming Theorem 6.5.3.
Proof. By Theorem 6.5.3, we have that T 2r(a)
p ≃ eH 1 p(Φ 2 , L n (Z p ))⊗T p . Hence, for all n ≡
a (mod 2), we have V 0 (a)[P n ] ≃ Tp 2r(a) . The theorem now follows from Lemma 6.5.2.
Now, we shall show that the inclusions and isomorphisms in (6.5.1) hold. This is the
content of the next few lemmas and propositions. The following proposition proves the
inclusion (1) in (6.5.1).
83

Proposition 6.5.4. For all r ≥ 1 and n ≥ 0, we have
eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /p r Z p ↩→ eH 1 p(Φ 2 , L n (Z p /p r Z p )).
Proof. For any Z p -module A, the short exact sequence of modules
0 → eH 1 p(Φ 2 , L n (Z p )) → eH 1 (Φ 2 , L n (Z p )) → eH 1 (Φ 2 , L n (Z p ))/eH 1 p(Φ 2 , L n (Z p )) → 0
induces the long exact sequence
Tor(M, A) → eH 1 p(Φ 2 , L n (Z p )) ⊗ A α → eH 1 (Φ 2 , L n (Z p )) ⊗ A,
where M = eH 1 (Φ 2 , L n (Z p ))/eH 1 p(Φ 2 , L n (Z p )). If M is Z p -free, then the map α is injective.
Now, for any congruence subgroup Φ, we have:
eH 1 (Φ, L n (Z p )) ⊗ A ∼ → eH 1 (Φ, L n (A)).
Under this identification, the map α preserves parabolic classes, proving the theorem.
The Z p -freeness of eH 1 (Φ 2 , L n (Z p ))/eH 1 p(Φ 2 , L n (Z p )) follows from Proposition 6.4.6.
The isomorphisms (2) and (3) in (6.5.1) follows from [Hid86a, Cor. 4.5], noting that
the argument given there works for p = 2 and for Φ 2 , instead of p odd and the Φ 1 there.
The following lemma proves the inclusion (4).
Lemma 6.5.5. For r ≥ 2, we have an inclusion
eH 1 p(Φ r , Z p /p r Z p (n)) ↩→ eH 1 p(Γ 1 (Np r ), n; Z p /p r Z p ).
Proof. We have the following inflation-restriction sequence for the groups Γ 1 (Np r ) ⊆ Φ r :
0 → H 1 (Φ r /Γ 1 (Np r ), Z p /p r Z p (n)) → H 1 (Φ r , Z p /p r Z p (n))
→ H 1 (Γ 1 (Np r ), Z p /p r Z p (n)) Φr/Γ 1(Np r) .
Since H 1 (Γ 1 (Np r ), Z p /p r Z p (n)) Φr/Γ 1(Np r) ↩→ H 1 (Γ 1 (Np r ), n; Z p /p r Z p ), we have the following
exact sequence
0 → H 1 (Φ r /Γ 1 (Np r ), Z p /p r Z p (n)) → H 1 (Φ r , Z p /p r Z p (n)) → H 1 (Γ 1 (Np r ), n; Z p /p r Z p ).
By [Hid86a, Lem. 6.1], we have the required claim.
The following lemma proves inclusion (5) in (6.5.1).
Lemma 6.5.6. eH 1 p(Γ 1 (Np r ), n; Z p /p r Z p ) ↩→ V 0 (a)[P n ], if n ≡ a (mod 2).
Proof. Observe that, eH 1 p(Γ 1 (Np r ), n; Z p /p r Z p ) is the subspace of eH 1 p(Γ 1 (Np r ), Z p /p r Z p )
on which Z × p act by v|z = z n v. The group V 0 (a)[P n ] is also the subspace of V 0 such that
Z × p acts by v|z = z n v, since µ 2 acts by ζ2 n = ζa 2 and γ ∈ Γ acts by v|γ = γn v.
By [Hid86a, p. 584, (5.4)], we have that eH 1 p(Γ 1 (Np r ), Z p /p t Z p ) ≃ eH 1 p(Γ 1 (Np r ), Z p )⊗
Z p /p t Z p . Since tensor product commutes with direct limits, we have that
lim eH
−→ 1 p(Γ 1 (Np r ), Z p /p t Z p ) ≃ eH 1 p(Γ 1 (Np r ), Z p ) ⊗ Zp T p ≃ eH 1 p(Γ 1 (Np r ), T p ).
t
By part (1) of Theorem 6.3.1 with V 0 , we have eH 1 p(Γ 1 (Np r ), Z p /p r Z p ) ↩→ V 0 . Since this
map respects the action of Z × p , we have the required claim.
84

6.6 Constant rank
In this section, we prove that the ranks of certain cuspidal ordinary 2-adic Hecke algebras
of different weights are all equal to the rank of a weight 2 cuspidal ordinary Hecke algebra.
As in the previous subsection, p = 2 and N > 1 is odd.
For a = 0 or 1, recall that r(a) is the rank of the Hecke algebra h 0 2 (Φ 2, ω a , Z p ), where
ω denotes the mod 4 cyclotomic character. For simplicity, we write A(ω n ) for the sheaf
with twisted action L 0 (ω n , A), for any Z p -module A.
Theorem 6.6.1. For each positive integer n ≡ a (mod 2),
rank Zp
h 0 n+2(Φ 2 , Z p ) = r(a).
Before proving this theorem, we need to gather some results, which we do now.
Lemma 6.6.2. For r > s ≥ 0, the module eH 1 (Φ s r, L n (Z p )) is Z p -free, for n ≥ 0.
Proof. The short exact sequence
0 → L n (Z p ) → L n (Q p ) → L n (T p ) → 0
induces a long exact sequence of cohomology groups for the group Φ s r
H 0 (Φ s r, L n (Q p )) α → H 0 (Φ s r, L n (T p )) β → H 1 (Φ s r, L n (Z p )) γ → H 1 (Φ s r, L n (Q p )).
If n = 0, then the map α is surjective, and hence the map β is zero. Therefore, the
map γ is injective, and hence H 1 (Φ s r, Z p ) is Z p -free. Assume that n > 0. If we can show
that eH 0 (Φ s r, L n (T p )) = 0, then the lemma follows. The operator T p acts on L n (T p )
by x|T p = ∑ p−1( 1 −i
) ι
i=0 0 p · x, where A ι = Adj(A). We see that T p acts on any element
of H 0 (Φ s r, L n (T p ))[p] by the matrix ( )
0 ∗
0 0 , and hence T
2
p acts trivially on such elements,
hence the idempotent e annihilates H 0 (Φ s r, L n (T p )).
Corollary 6.6.3. For any integer n ≥ 0, the module eH 1 (Φ 2 , Z p (ω n )) is Z p -free.
Proof. If n is even, then ω n = 1, hence this follows from the lemma and when n is odd,
the proof is similar to the proof of the lemma.
Lemma 6.6.4. eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /qZ p ≃ eH 1 p(Φ 2 , L n (Z p /qZ p )).
Proof. By Proposition 6.5.4 with r = 2, the map
eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /qZ p → eH 1 p(Φ 2 , L n (Z p /qZ p ))
is injective. For the surjectivity, we work with sheaf cohomology instead of group cohomology.
Let Y be the complex open manifold associated with Φ 2 . Observe that we have
the following commutative diagram:
eH 1 c(Y, F (L n (Z p )))/q
eH 1 c(Y, F (L n (Z p /qZ p )))
≀
eH 1 p(Y, F (L n (Z p )))/q
eH 1 p(Y, F (L n (Z p /qZ p )))
eH 1 (Y, F (L n (Z p )))/q
eH 1 (Y, F (L n (Z p /qZ p ))).
By [Hid88b, Cor. 2.2], the first vertical map is an isomorphism. As a result, we get that
the middle vertical map is surjective and the lemma follows.
≀
85

Lemma 6.6.5. For any n ≥ 0, eH 1 (Y, F (Z p (ω n )))/q ≃ eH 1 (Y, F (Z p /qZ p (ω n ))).
Proof. The short exact sequence
induces another short exact sequence
0 → Z p (ω n ) q → Z p (ω n ) → Z p /qZ p (ω n ) → 0
0 → H 1 (Y, F (Z p (ω n ))) ⊗ Z p /qZ p → H 1 (Y, F (Z p /qZ p (ω n ))) → H 2 (Y, F (Z p (ω n )))[q] → 0,
where H 2 (Y, F (Z p (ω n )))[q] = {x ∈ H 2 (Y, F (Z p (ω n ))) | q·x = 0}. This last group vanishes,
since the cohomological dimension of Φ s r is 1.
Proposition 6.6.6. The module e(H 1 (Φ 2 , Z p (ω))/H 1 p(Φ 2 , Z p (ω))) is Z p -free.
Proof. If the module eG 1 (Φ 2 , Z p (ω)) is Z p -free, then the proposition follows, since the
maps in (6.1.1) are invariant under the action of the Hecke operators. Consider the long
exact sequence of cohomology groups
G 0 (Φ 2 , Q p (ω)) β → G 0 (Φ 2 , T p (ω)) → G 1 (Φ 2 , Z p (ω)) → G 1 (Φ 2 , Q p (ω)),
induced by the short exact sequence 0 → Z p (ω) → Q p (ω) → T p (ω) → 0.
Since the image of β is p-divisible, it is enough to show that, ∀ x ∈ G 0 (Φ 2 , T p (ω))[p],
x|T p belongs to β(G 0 (Φ 2 , Q p (ω))). Then a small computation shows that
e(G 0 (Φ 2 , T p (ω))/β(G 0 (Φ 2 , Q p (ω)))) = 0,
and hence eG 1 (Φ 2 , L n (Z p (ω))) is Z p -free. We now prove, for all x ∈ G 0 (Φ 2 , T p (ω))[p], the
element x|T p belongs to β(G 0 (Φ 2 , Q p (ω))).
Since (2, N) = 1 and N ≥ 3, all the cusps of Φ 2 are regular, because irregularity for
Φ 2 implies the irregularity for Γ 1 (N), but there are no irregular cusps for Γ 1 (N). Let
s ∈ C(Φ 2 ) be a cusp of Φ 2 . Let α s = ( )
a b
c d ∈ SL2 (Z) such that α s (∞) = s. If π s denotes
a generator for (Φ 2 ) s , we may write
(
π s = α 1 u
)
s 0 1 α
−1
s = ( )
1−cau a 2 u
−c 2 u 1+cau ∈ Φ2 , with u ≠ 0. (6.6.1)
The structure of G 0 (Φ 2 , M(ω)) depends on the action π s on M. In order to study
this, let us divide the cusps into two types. If p | u, then we refer to this cusp as being of
type 1, otherwise of type 2. We assume that Φ 2 acts trivially on M, because we are only
interested in the cases when M = Z p , Q p or T p . Let x be an element of G 0 (Φ 2 , T p (ω))[p]
and let x s denote the component of x in H i ((Φ 2 ) s , T p (ω)).
If s is a cusp of type 1, then we see that H 0 ((Φ 2 ) s , M(ω)) = M by (6.6.1) and
moreover the map β s is surjective, where β s : H 0 ((Φ 2 ) s , Q p (ω)) → H 0 ((Φ 2 ) s , T p (ω)).
Hence (x|T p ) s ∈ β(H 0 ((Φ 2 ) s , Q p (ω)) = Q p ).
Suppose s is a cusp of type 2. If π s acts trivially on M, then H 0 ((Φ 2 ) s , M(ω)) = M
and if π s does not act trivially on M, then H 0 ((Φ 2 ) s , M(ω)) = M[2]. In the former case,
again (x|T p ) s ∈ β(Q p ). In the latter case,
p−1
∑( (
(x|T p ) s = γ 1 0
) )
0 p π
i ι
s · xt ,
i=0
86

where γ ∈ Φ 2 such that t = γ ( )
1 0
0 p (s) ∈ C(Φ2 ). Since x is 2-torsion, we see that
(x|T p ) s = ∑ p−1
i=0 (±1) · x t = ∑ p−1
i=0 x t = 2x t = 0 ∈ β(H 0 ((Φ 2 ) s , Q p (ω))) = 0. Hence we have
that for any x ∈ G 0 (Φ 2 , T p (ω))[p], the element x|T p belongs to β(G 0 (Φ 2 , Q p (ω))).
Remark 6.6.7. In the above proof, we have used the fact that p is 2.
Corollary 6.6.8. eH 1 p(Φ 2 , Z p (ω n ))/q ↩→ eH 1 p(Φ 2 , Z p /qZ p (ω n )).
Proof. When n is even, this follows from Proposition 6.5.4 with r = 2. When n is odd,
the injectivity of the first vertical map follows from the following diagram
eH 1 p(Y, F (Z p (ω)))/q α
eH 1 (Y, F (Z p (ω)))/q
eH 1 p(Y, F (Z p /qZ p (ω)))
eH 1 (Y, F (Z p /qZ p (ω))),
≀
since α is injective by Proposition 6.6.6, and the second vertical map is an isomorphism
by Lemma 6.6.5.
Now we shall give a proof Theorem 6.6.1.
Proof. It is enough to prove that the Z p -rank of eH 1 p(Φ 2 , L n (Z p )) is the same as the Z p -
rank of eH 1 p(Φ 2 , Z p (ω a )) (the modules are Z p -free by Lemma 6.6.2 and by its corollary).
By (6.5.2), we see that the rank of eH 1 p(Φ 2 , L n (Z p )) is less than or equal to the rank of
eH 1 p(Φ 2 , Z p (ω a )).
Again by Lemma 6.6.2 and by its corollary, it is enough to show the Z p /qZ p -rank of the
module eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /qZ p is greater than or equal to that of eH 1 p(Φ 2 , Z p (ω a )) ⊗
Z p /qZ p . We have the following
eH 1 p(Φ 2 , L n (Z p )) ⊗ Z p /qZ p ≃
(1)
eH 1 p(Φ 2 , L n (Z p /qZ p ))
≃
(2) eH1 p(Φ 2 , Z p /qZ p (ω a ))
←↪
(3) eH1 p(Φ 2 , Z p (ω a )) ⊗ Z p /qZ p ,
(6.6.2)
where the isomorphisms (1), (2) and the inclusion (3) follow from Lemma 6.6.4, the
isomorphism (3) in (6.5.1) with r = 2, and Corollary 6.6.8, respectively. Hence the
theorem is proved.
6.7 Λ-adic Hecke algebras
Recall that our aim is to prove a control theorem for Hida’s ordinary Hecke algebra, which
we now introduce.
For each subalgebra A of C or C p , and for any r ≥ s ≥ 1, we have a commutative
diagram for all n:
S k (Γ 1 (Np s ), A) S k (Γ 1 (Np r ), A)
T n
S k (Γ 1 (Np s ), A)
T n
S k (Γ 1 (Np r ), A).
87

Thus, we have a A-algebra homomorphism, h k (Γ 1 (Np r ), A) ↠ h k (Γ 1 (Np s ), A) and since
T p ↦→ T p , we have that h 0 k (Γ 1(Np r ), A) → h 0 k (Γ 1(Np s ), A), for each r ≥ s ≥ 1. Now, set:
h k (Γ 1 (Np ∞ ), A) := lim h ←−r k (Γ 1 (Np r ), A), h 0 k (Γ 1(Np ∞ ), A) := lim h 0 ←−r k (Γ 1(Np r ), A),
S k (Np ∞ , A) := ∪ ∞ r=1S k (Γ 1 (Np r ), A).
In Lemma 6.7.2 below, we show that, for weights k 1 ≥ k 2 ≥ 2, there is a surjection
h k1 (Γ 1 (Np ∞ ), A) ↠ h k2 (Γ 1 (Np ∞ ), A), and hence on the ordinary parts. Before we state
it, we need to define a pairing between certain Hecke algebras and certain spaces of
modular forms. Recall that K/Q p is finite, and O K is the integral closure of Z p in K.
Put
S k (Np r , K/O K ) = S k (Γ 1 (Np r ), K)/S k (Γ 1 (Np r ), O K ).
By definition, one can embed this space via q-expansion into the module of formal series
K/O K [[q]]. We take the injective limit:
S k (Np ∞ , K/O K ) = lim S −→r k (Np r , K/O K ) → K/O K [[q]].
Then S k (Np ∞ , K/O K ) ≃ S k (Np ∞ , K)/S k (Np ∞ , O K ). The algebra h k (Γ 1 (Np ∞ ), O K )
acts on S k (Np ∞ , K/O K ). Define the pairing
(, ) : h k (Γ 1 (Np ∞ ), O K ) × S k (Np ∞ , K/O K ) → K/O K ,
by (h, f) = a(1, f|h). Then (h, f|g) = (hg, f), for all h, g ∈ h k (Γ 1 (Np ∞ ), O K ). Equip
S k (Np ∞ , K/O K ) with the discrete topology. We have (cf. [Hid86a, Lem. 7.1]):
Lemma 6.7.1. The pairing above shows that h k (Γ 1 (Np r ), Z p ) and S k (Γ 1 (Np r ), T p ) (respectively,
h 0 k (Γ 1(Np r ), Z p ) and S 0 k (Γ 1(Np r ), T p )), for r = 1, 2, . . . , ∞, are Pontryagin
duals.
Lemma 6.7.2. For k 1 ≥ k 2 ≥ 2, there exists a surjection
h k1 (Γ 1 (Np ∞ ), O K ) ↠ h k2 (Γ 1 (Np ∞ ), O K ).
Proof. The proof is similar to the proof of [Hid86a, Lem. 7.2]. For p = 2, we need to work
with a different Eisenstein series than the one given in that lemma. For r ≥ 2, define a
formal q-expansion for each t ∈ (Z/p r Z) × by
⎛
⎞
G(r, t) = −t 0 p −r + 1 ∞ 2 + ∑ ∑
⎝
sgn(d) ⎠ q n ,
n=1
d|n, d≡t (mod p r )
where t 0 is an integer satisfying 0 ≤ t 0 < p r and t 0 ≡ t mod p r . Then, as shown by Hecke,
G(r, t) gives the q-expansion of an element of M 1 (Γ 1 (Np r ), Q) and satisfies
G(r, t)| 1 = G(r, at) for ( a b
c d
)
∈ Γ0 (Np r ).
Put E(r, t) = −p r G(r, t). For odd primes p, the congruence E(r, t) ≡ t (mod p r ) holds.
For the prime p = 2, the congruence that holds is E(r, t) ≡ t (mod p r−1 ). Multiplication
by the Eisenstein series E(r, 1) gives an injective morphism
ι r : S k−1 (Np ∞ , T p )[p r−1 ] → S k (Np ∞ , T p )[p r−1 ].
Using the injective limit of the maps ι r and Lemma 6.7.1, we can finish the proof of the
lemma along the lines of the proof of [Hid86a, Lem. 7.2].
88

It is known, by [Hid88b, Thm. 3.2], that the map in the lemma above is an isomorphism.
This theorem is stated adelically, but includes the case of p = 2. Thus, the Hecke
algebra h 0 k (Γ 1(Np ∞ ), O K ) is independent of the weight, for all k ≥ 2, and we denote this
by h 0 (N, O K ).
6.8 Control theorem for ordinary Hecke algebras
In this section, we prove a control theorem for Hida’s ordinary Hecke algebras for the
prime p = 2.
Recall that K is a finite extension of Q p and O K is integral closure of Z p in K. Let ɛ
be a character of Γ/Γ r with values in O K , with r ≥ 2. In this section, we write Λ for Λ K
and Q(Λ) for the field of fractions of Λ K .
We know that h 0 (N, O K ) acts on the finite free Λ-module V 0 . Hence the Λ-module
h 0 (N, O K ) is finitely generated and torsion-free, since the action is faithful on V 0 . By
abuse of notation, let P k,ɛ also denote the prime ideal generated by the prime element
P k,ɛ = ι(u) − ɛ(u)u k . By the independence of weight of h 0 (N, O K ), there is a surjective
homomorphisms of O K -algebras, respectively, of Λ-algebras:
ρ : h 0 (N, O K ) ↠ h 0 k (Φ2 r, ɛ, O K ) and Λ Pk,ɛ ↠ Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ = K, (6.8.1)
where Λ Pk,ɛ /P k,ɛ Λ Pk,ɛ is identified with K with ι(u) corresponding to u k ɛ(u), inducing the
map
˜ρ k,ɛ : h 0 (N, O K ) ⊗ Λ Λ Pk,ɛ ↠ h 0 k (Φ2 r, ɛ, O K ) ⊗ OK K,
which in turn factors via P k,ɛ to give the map:
Theorem 6.8.1. The natural map
is an isomorphism.
ρ k,ɛ : h 0 (N, O K ) ⊗ Λ Λ Pk,ɛ /P k,ɛ ↠ h 0 k (Φ2 r, ɛ, K).
ρ k,ɛ : h 0 (N, O K ) ⊗ Λ Λ Pk,ɛ /P k,ɛ ↠ h 0 k (Φ2 r, ɛ, K)
Proof. Since the module h 0 (N, O K ) is finitely generated torsion-free over Λ, then so is
h 0 (N, O K ) Pk,ɛ over Λ Pk,ɛ . Therefore, the module h 0 (N, O K ) Pk,ɛ is free, since any finitely
generated torsion-free module over a discrete valuation ring is free. Let S(k, ɛ) (resp.,
R(k, ɛ)) denote the rank of h 0 (N, O K ) Pk,ɛ (resp., of h 0 k (Φ2 r, ɛ, K)). A priori, the number
S(k, ɛ) depends on k and ɛ. Since
h 0 (N, O K ) Pk,ɛ ⊗ ΛPk,ɛ Q(Λ) ≃ h 0 (N, O K ) ⊗ Λ Q(Λ),
we see that S(k, ɛ) is independent of k and ɛ and we denote this common value by R.
We first prove the theorem for weights k > 2, by assuming that it holds for k = 2.
The Eisenstein series E(2, 1) (cf. §6.7) has the property that E(2, 1) ≡ 1 (mod 2).
Multiplication by E(2, 1) k−2 induces an injection
S 0 2(Γ 1 (Np r ), T p )[p] → S 0 k (Γ 1(Np r ), T p )[p].
89

By duality, we have a surjection
h 0 k (Γ 1(Np r ), Z p ) ⊗ Z p /pZ p ↠ h 0 2(Γ 1 (Np r ), Z p ) ⊗ Z p /pZ p .
Then
R[Γ : Γ r ] ≥ ∑ ɛ
R(k, ɛ) = rank OK (h 0 k (Γ 1(Np r ), O K ))
≥ rank OK (h 0 2(Γ 1 (Np r ), O K )) = R[Γ : Γ r ],
where the last equality follows by assumption. This can happen only if R(k, ɛ) = R for
all k, ɛ, showing that the map ρ k,ɛ is an isomorphism.
Now, we shall prove the result for k = 2. By Theorem 6.3.1, we have that the Z p -rank
of eH 1 p(Γ 1 (Np r ), Z p ) is equal to 2[Γ : Γ r ] rank Zp h 0 2 (Γ 1(Nq), Z p ). Hence,
rank Zp h 0 2(Γ 1 (Np r ), Z p ) = [Γ : Γ r ] rank Zp h 0 2(Γ 1 (Nq), Z p ).
Since h 0 2 (Γ 1(Np r ), K) = ⊕ ɛ h 0 2 (Φ2 r, ɛ, K), the left hand side of the equality above is also
∑
ɛ R(2, ɛ). If rank Z p
h 0 2 (Γ 1(Nq), Z p ) = R, then [Γ : Γ r ]R = ∑ ɛ
R(2, ɛ). Since R ≥
R(2, ɛ), we get R = R(2, ɛ), for each ɛ, as desired. Thus, we need to show that R =
rank Zp h 0 2 (Γ 1(Nq), Z p ). This is proved in Theorem 6.8.3 below.
The following lemma is well-known; for the proof refer to [Hid86a, Lem. 6.4].
Lemma 6.8.2. For any subfield K of C or C p , H 1 p(Γ 1 (M), L n (K)) is free of rank 2 over
the Hecke algebra h n+2 (Γ 1 (M), K) for each positive integer M.
Set ɛ := ( )
1 0
0 −1 . Let M be a module over the Hecke algebra h
0
2 (Γ 1 (Np r ), Z). Let
M ± denote the subspaces of M defined by {m ± [ɛ]m | m ∈ M}. Since ɛ normalizes
Γ 1 (Np r ), the action of [ɛ] = [Γ 1 (Np r )ɛΓ 1 (Np r )] commutes with that of h 0 2 (Γ 1(Np r ), Z)
on M. Therefore, the modules M ± are stable under the action of h 0 2 (Γ 1(Np r ), Z). For
simplicity, we write h 0 2 (N, Z p) for the weight-2 Λ-adic Hecke algebra h 0 2 (Γ 1(Np ∞ ), Z p ).
Theorem 6.8.3. The surjective map
is an isomorphism.
ρ 2,triv : h 0 2(N, Z p ) ⊗ Λ Λ P2 /P 2 ↠ h 0 2(Γ 1 (Nq), Q p )
Proof. By Theorem 6.3.1, we have (V 0 ) Γ 2
= V 0 2 , i.e., V0 [P 2 ] = V 0 [ω 2,2 ] = eH 1 (X 2 , T p ) =
eH 1 p(Γ 1 (Nq), T p ), where the last equality follows from [Hid86a, p. 583 (5.3)]. Again by
the same theorem, we have
V 0 /P 2 V 0 ≃ Hom Zp (eH 1 p(Γ 1 (Nq), Z p ), Z p ). (6.8.2)
Since V 0 is direct limit over V 0 r , we see that [ɛ] acts on V 0 and the action commutes with
that of h 0 2 (N, Z p). There is a map V 0+ ⊕ V 0− → V 0 , which is an isomorphism if p is odd.
Since p = 2, we tensor this with Λ P2 to get an isomorphism (V 0+ ) P2 ⊕ (V 0− ) P2 ≃ V 0 P 2
.
Let V 0± denote the Pontryagin dual of V 0± . Then (V 0+ ) P2 ⊕ (V 0− ) P2 ≃ V 0 P 2
. We can
think of h 0 2 (N, Z p) P2 as a subalgebra of the endomorphism algebra of (V 0+ ) P2 and hence
we shall restrict ourselves to the module (V 0+ ) P2 . We now prove that
(V 0+ ) P2 /P 2 (V 0+ ) P2 = h 0 2(Γ 1 (Nq), Q p ).
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We remark that since p = 2 and we work with (V 0+ ) P2 , the above isomorphism is with
Q p -coefficients, otherwise we would have worked with V 0+ and the above isomorphism
would have been with Z p -coefficients. Since the functor Hom Zp (−, T p ) commutes with
the ±-action after tensoring with Λ P2 , we see that V 0± ⊗ Λ Λ P2 ≃ (VP 0 2
) ± holds. Hence
(V 0± ) P2 /P 2 (V 0± ) P2 ≃ (VP 0 2
) ± /P 2 (VP 0 2
) ± ≃ (VP 0 2
/P 2 VP 0 2
) ± , where the last isomorphism is
an easy check. We have that
(V 0+ ) P2 /P 2 (V 0+ ) P2 = (V 0 P 2
/P 2 V 0 P 2
) +
= (Hom Z p
(eH 1 p(Γ 1 (Nq), Z p ), Z p ) ⊗ Zp Q p ) +
(6.8.2)
= Hom Qp (eH 1 p(Γ 1 (Nq), Q p ) + , Q p ) = h 0 2(Γ 1 (Nq), Q p ),
(6.8.3)
where the last equality follows from Lemma 6.8.2 and the third equality follows from the
fact that for any Q p -module M, Hom Qp (M, Q p ) ± ≃ Hom Qp (M ± , Q p ).
Let v denote the vector in (V 0+ ) P2 corresponding to 1 in h 0 2 (Γ 1(Nq), Q p ) in (6.8.3).
Therefore, we have a map h 0 2 (N, Z p) P2 → (V 0+ ) P2 defined by mapping h → hv. This
map is a surjective map by Nakayama’s lemma and by (6.8.3). The map is injective
since the Hecke action is faithful on (V 0+ ) P2 . Therefore, we have h 0 2 (N, Z p) P2 ≃ (V 0+ ) P2 .
Tensoring this isomorphism with Λ P2 /P 2 and using (6.8.3), we obtain the theorem.
6.9 Uniqueness
In this section, we prove a uniqueness result for Hida families. Suppose f is a p-stabilized
ordinary newform (or an eigenform). Let P f denote the unique height one prime ideal,
induced by f, via the isomorphism in Theorem 6.8.1. Suppose Q = P f lies over the prime
ideal P = P k,ɛ , where the integer k and the character ɛ depend on f.
First we show that, for the prime P of Λ, the localized Hecke algebra h 0 (N, O K ) Q is
étale over Λ Pk,ɛ . We deduce the uniqueness result as a consequence. For simplicity, let us
denote h 0 (N, O K ) by h 0 (N).
Proposition 6.9.1. The localized Hecke algebra h 0 (N) Q is étale over Λ P and Qh 0 (N) Q =
P h 0 (N) Q , i.e., h 0 (N) Q is a regular local ring.
Proof. We apply [Nek06, Lem. 12.7.6], with A = Λ, B = h 0 (N) and J = 0 (and also by
switching the roles of P and Q). The first condition of that lemma, namely the Hecke
algebra h 0 (N) is finitely generated and torsion-free over Λ follows, as mentioned earlier,
from Theorem 6.3.1. By Theorem 6.8.1, the short exact sequence in the second part of
that lemma reduces to
0 → P → h 0 k (Φ2 r, ɛ, K) α → Q p (a n (f)) ∞ n=1 → 0,
where the map α is given by T n → a n (f). By analyzing the proof of that lemma, we see
that if P P = 0, where P P denotes the localisation, then the proposition follows. From the
theory of newforms, one knows that h 0 k (Φ2 r, ɛ, K) P
∼ → Qp (a n (f)) ∞ n=1 , hence P P = 0.
Remark 6.9.2. The étaleness of h 0 (N) new
P k,ɛ
over Λ Pk,ɛ for the N-new part of Hida’s Hecke
algebra would follow from a control theorem for h 0 (N) new instead of h 0 (N) if one knew
such a result. To prove the étaleness, it is enough to show Ω h 0 (N) new
P /Λ P = 0. Let K P be the
91

esidue field of Λ P . By Nakayama’s lemma it suffices to show that Ω h 0 (N) new
P /Λ ⊗ P Λ P
K P =
Ω h 0 (N) new
P ⊗K P /K P
= 0. The vanishing of the Kähler differentials follows from the fact that
P is an arithmetic point and by the control theorem with “h 0 (N) new ”, we have that
h 0 (N) new
P
⊗ K P is a classical new Hecke algebra acting on the space of classical newforms
of fixed level, weight and character. Such Hecke algebras are well-known to be semisimple,
since they are generated by Hecke operators away from Np.
6.9.1 2-adic Λ-adic forms
Now we recall the definition of a 2-adic Λ = Z 2 [[X]]-adic form. Let I denote the integral
closure of Λ in a finite extension L of Q(Λ). Let ζ denote a 2 r−2 -th root of unity in ¯Q 2 , with
r ≥ 2, and let k ≥ 1 be a positive integer. The assignment X ↦→ ζ(1 + q) k − 1 yields a Z 2 -
algebra homomorphism ϕ k,ζ : Λ → ¯Q 2 . We say that a height one prime P ∈ Spec(I)( ¯Q 2 )
has weight k, if the corresponding Λ-algebra homomorphism P : I → ¯Q 2 extends ϕ k,ζ on
Λ, for some k ≥ 1 and for some ζ. In addition, we say that P is arithmetic, if P has
weight k ≥ 2.
From now on, p = 2, q = 4 and (2, N) = 1. We will need the following Dirichlet
characters. Let
• ψ be a Dirichlet character of level Nq,
• ω be the mod 4 cyclotomic character,
• ɛ be the character χ ζ mod 2 r for each root of unity ζ of order 2 r−2 with r ≥ 2
defined by first decomposing
(Z p /2 r Z p ) × = (Z p /qZ p ) × × Z/2 r−2 ,
where the second factor is generated by 1 + q, and then by setting
χ ζ = 1 on (Z p /qZ p ) × and χ ζ (1 + q) = ζ.
Definition 6.9.3. Let F = ∑ ∞
n=1 a(n, F)qn ∈ I[[q]] be a formal q-expansion. We say F is
a Λ-adic form of tame level N and character ψ if for each arithmetic point P ∈ Spec(I)( ¯Q p )
lying over ϕ k,ζ , with k ≥ 2 and ζ of order p r−2 , r ≥ 2, the specialization
P (F) ∈ ¯Q p [[q]]
of F at P is the q-expansion of a classical cusp form f ∈ S k (N2 r , χ), where χ = ψω −k χ ζ .
Now, we briefly recall various notions, e.g., primitive, ordinary, for Λ-adic forms.
Definition 6.9.4. We say that F is a Λ-adic eigenform of level N, if every arithmetic
specialization f = P (F) is an eigenvector for the classical Hecke operators T l for all
primes l with l ∤ Np, and for U l if l | Np.
Let f ∈ S k (Np ∞ , χ) be any eigenvector for all the operators T l with l ∤ Np. Atkin-
Lehner theory implies that one can associate to f a unique primitive form f ∗ of minimal
level dividing Np ∞ which has the same eigenvalues as f for almost all primes l.
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Definition 6.9.5. We say that a Λ-adic eigenform F of level N is a newform of level N,
if for every arithmetic specialization f = P (F) ∈ S k (Np ∞ , χ), the associated primitive
form f ∗ has level divisible by N.
We say that F is normalized, if a(1, F) = 1. Finally, let us say that F is primitive
of level N, if it is normalized and is a newform of level N.
There is a well-known theory of Λ-adic forms due to Hida under the assumption of
ordinariness, which we describe now. Let f be a normalized eigenform for all the Hecke
operators of level Np r , with (N, p) = 1 and r ≥ 2, and weight k ≥ 2. Assume that f ∗
has level divisible by N. We say that f is ℘-stabilized (or p-stabilized), if either f is
p-new and is ℘-ordinary, or f is p-old, and is obtained from a primitive ℘-ordinary form
f 0 of level N by the formula f(z) = f 0 (z) − βf 0 (pz), where β is the non ℘-adic-unit root
of E p (x) = x 2 − a(p, f 0 )x + χ(p)p k−1 . The cusp form f is an eigenform of all the Hecke
operators of level Np, with U p eigenvalue equal to α, the unique ℘-adic unit root of E p (x).
Definition 6.9.6. We say that a primitive Λ-adic form of level N is ℘-ordinary, if each
arithmetic specialization P (F ) is a ℘-stabilized form in S k (N, χ), where k, N, χ are the
weight, level, and character of P respectively.
6.9.2 Tame level N = 1
We remark that there is no 2-adic Hida theory, when N = 1, showing that our assumption
that N > 1 in several previous sections loses no generality. Indeed, we have that:
Proposition 6.9.7. There are no ordinary Λ-adic eigenforms of tame level 1.
Proof. If such a Λ-adic eigenform were to exist, then for every integer k ≥ 2 and r ≥ 2,
its’ specialization at P k,ζ , where ζ is a p r−2 -th root of unity, would be an element of
S k (p r , ω a−k χ ζ ), for some a ∈ N. For parity reasons, (−1) a−k = (−1) k , hence a is even.
But, if r = 2 and k is even, then are no 2-ordinary, 2-stabilized eigenforms in S k (4, triv).
For newforms this follows from [Miy89, Thm. 4.6.17] and for oldforms from loc. cit. and
the fact that X 0 (2) has genus 0, and from Hatada [Hat79]. Since there are no ordinary
specializations in even weight, there are no ordinary Λ-adic eigenforms of tame level 1.
Corollary 6.9.8. For odd integers k, the space eSk
2-new (4, ω) is zero.
Proof. This follows immediately from the proposition noting that every 2-ordinary eigenform
in the above space must live in a 2-ordinary Hida family of tame level 1.
Remark 6.9.9. It can be checked independently that the dimensions of eSk
2-new (4, ω) for
k = 3, 5, 7, 9, 11, 13, 15, 17, are indeed all zero, whereas the dimensions of Sk 2-new (4, ω)
for k = 3, 5, 7, 9, 11, 13, 15, 17 are 0, 1, 2, . . . , 7, respectively.
6.9.3 Uniqueness result
We now turn to the uniqueness result for 2-adic families.
Theorem 6.9.10. Any p-stabilized eigenform is an arithmetic specialization of a unique
Hida family, up to Galois conjugacy.
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Proof. By (6.8.1), we know that any p-stabilized eigenform lives in a Hida family. We
want to show that such a family is unique, up to Galois conjugacy (this last caveat is
necessary since if a form lies in F by specialization under P : I → ¯Q p , then it also lies in
the conjugate family F σ , by specializing under σ −1 ◦ P : I σ → ¯Q p . Note that F and F σ
correspond to the same minimal prime ideal of h 0 (N)).
Assume the contrary. Let λ 1 and λ 2 denote the algebra homomorphisms from h 0 (N)
to I and I ′ respectively, where I, I ′ are finite integral extensions of Λ. Let P 1 and
P 2 denote the minimal prime ideals of h 0 (N) which are the respective kernels of these
homomorphisms. Since λ 1 and λ 2 have one arithmetic specialization in common, there
are two algebra homomorphisms P : I → ¯Q p and P ′ : I ′ → ¯Q p such that P ◦ λ 1 =
P ′ ◦ λ 2 = λ P,P ′, say. Then the kernel of λ P,P ′ is a height one prime of h 0 (N), denote by
Q, containing both P 1 and P 2 and lying over P = P k,ζ for some k ≥ 2, ζ.
By Proposition 6.9.1, h 0 (N) Q is a regular local ring. But, a regular a local ring is a
domain, hence the prime ideals P 1 and P 2 have to be equal.
As an application of the last result we now show that the notion of CM-ness is pure
with respect to families.
Proposition 6.9.11. Let F be a primitive 2-adic Hida family. Then either all arithmetic
specializations are CM forms or no arithmetic specialization is a CM form.
Proof. The proof is the same as for odd prime p, once one has the uniqueness result for
p = 2. Indeed a CM family is defined to be one which is obtained as the theta series
of a Λ-adic Hecke character of an imaginary quadratic field. Clearly all its arithmetic
specializations are CM forms. Now start with an arbitrary CM form. Assume it lives in
a non-CM family (one which is not a theta series). Then explicit interpolation allows us
to also construct a CM family passing through this CM form. Clearly the non-CM family
and the CM family are not Galois conjugate, and this contradicts uniqueness.
In view of this result from now on we may and do speak of CM and non-CM 2-adic
Hida families.
6.10 An application to Galois representations
In [GV04], the splitting of the local Galois representations associated to ordinary eigenforms
was studied for odd primes p. We carry out the same analysis for the case of p = 2,
assuming that the relevant result of Buzzard continues to hold for p = 2 in the residually
dihedral setting. That is, under this assumption, we prove that in a non-CM 2-adic Hida
family, all arithmetic specializations have non-split local Galois representation, except
for a possible finite set of exceptions. As a consequence of the uniqueness result proved
earlier, we are also able to exclude CM forms from this finite exceptional set.
Recall p denotes the prime 2 and q = 4. We recall some preliminaries on ordinary
eigenforms and their associated Galois representations.
Let f = ∑ ∞
n=1 a n(f)q n be a normalized eigenform (i.e., a primitive form) of weight
k ≥ 2, level Np r ≥ 1, and nebentypus χ. Let ℘ be the prime of ¯Q determined by a fixed
embedding of ¯Q into ¯Q p . Let ℘ also denote the induced prime of K f = Q(a n (f)), the
94

Hecke field of f, and let K f,℘ denote the completion of K f at ℘. There is a global Galois
representation
ρ f = ρ f,℘ : Gal( ¯Q/Q) → GL 2 (K f,℘ ),
associated to f (and ℘) which has the property that for all primes l ∤ Np,
trace(ρ f (Frob l )) = a l (f) and det(ρ f (Frob l )) = χ(l)l k−1 .
Recall f is ordinary at ℘ (or ℘-ordinary), if a p (f) is ℘-adic unit. If f is ordinary at ℘,
then the result of Wiles [Wil88] shows that the restriction of ρ f to the decomposition
group G p is upper triangular, i.e.,
( )
δ ψ
ρ f | Gp ∼ ,
0 ɛ
where δ, ɛ : G p → K × f,℘ are characters with ɛ unramified and ψ : G p → K f,℘ is a continuous
function.
Definition 6.10.1. We say that the ordinary representation ρ f | Gp splits, if the representation
space of ρ f can be written as direct sum of two G p -invariant lines.
6.10.1 Buzzard’s result
We shall assume that a slight strengthening of a result of Buzzard holds. Let O denote
the ring of integers in a finite extension K of Q p , and λ denote the maximal ideal of O.
Let ρ : G Q → GL 2 (O) be a continuous representation and let ¯ρ denote the reduction mod
λ. The following result is proved in [Buz03], and we refer to that paper for a detailed
explanation of all the hypotheses.
Theorem 6.10.2 (Buzzard). Assume that
1. ρ is ramified at finitely many primes and ¯ρ is modular,
2. ¯ρ is absolutely irreducible when restricted to Gal( ¯Q/Q(i)),
3. ρ| Gp is the direct sum of two 1-dimensional characters α and β : G p → O x , such
that α(I p ) and β(I p ) are finite, and (α/β) mod λ is non-trivial,
4. ¯ρ(c) ≠ 1,
5. ¯ρ is both α-modular and β-modular, in the sense that there are eigenforms f α with
T p -eigenvalue ᾱ(Frob p ) and f β with T p -eigenvalue ¯β(Frob p ) giving rise to ¯ρ,
6. The projective image of ¯ρ is not dihedral.
Then ρ is modular, in the sense that there exists an embedding i : K ↩→ C and a
classical weight 1 cuspidal eigenform f such that the composite i ◦ ρ is isomorphic to the
representation associated to f by Deligne and Serre.
For odd primes p, there is no restriction (6) on the projective image of ¯ρ. For p = 2,
this assumption is due to the unavailability of R = T theorems in the residually dihedral
setting. From now on, we assume that Theorem 6.10.2 holds, without condition (6).
95

6.10.2 Λ-adic Galois representations
Now, we recall a few facts about Λ-adic Galois representations. Let F ∈ I[[q]] be a
primitive Λ-adic form of level N and with character ψ. There is a Galois representation
attached to F, constructed by Hida, and Wiles in the case of p = 2,
ρ F : G Q → GL 2 (L),
such that for each arithmetic specialization P of I, P (ρ F ), the specialization of ρ F at P ,
is isomorphic to the representation ρ f attached to f = P (F) by Deligne. Note that, if l
is a prime number such that l ∤ Np, then
trace(ρ F (Frob l )) = a(l, F) ∈ I, det(ρ F (Frob l )) = ψ(l)κ(Frob l )l −1 ,
where κ : Gal( ¯Q/Q) → Λ × is the ‘Λ-adic cyclotomic character’.
If F is ordinary, then the restriction of ρ F to G p turns out to be ‘upper-triangular’.
More precisely, the representation ρ F | Gp has the following shape
( )
δ F u F
ρ F | Gp ∼
,
0 ɛ F
where δ F , ɛ F : G p → L × are characters with ɛ F unramified, and u F : G p → L is a
continuous map. Let
c F = ɛ −1
F · u F ∈ Z 1 (G p , L(δ F ɛ −1
F
)) (6.10.1)
be the associated cocycle. Then the representation
ρ F | Gp
splits if and only if [c F ] = 0 in H 1 (G p , L(δ F ɛ −1
F )).
Definition 6.10.3. We say that F is p-distinguished, if the characters δ F and ɛ F appearing
above, have distinct reductions modulo the maximal ideal of I.
It follows that F is p-distinguished if and only if f is p-distinguished for one (therefore
every) arithmetic specialization f of F.
We shall show that for a primitive 2-adic family F whose residual representation satisfies
some technical conditions (cf. conditions (1), (2), and (3) below), the corresponding
representation ρ F splits at p if and if F is CM. As a consequence, standard descent arguments
(cf. §6.10.4) allow us to conclude the following partial result towards Greenberg’s
question on the local splitting of ordinary 2-adic modular Galois representations.
Theorem 6.10.4. Let F be a primitive non-CM 2-ordinary Hida family of eigenforms
with the property that
1. ¯ρ F is p-distinguished,
2. ¯ρ F is absolutely irreducible, when restricted to Gal( ¯Q/Q(i)),
3. ¯ρ F (c) ≠ 1 and ¯ρ F is both α-modular and β-modular.
Then for all but except possibly finitely many arithmetic members f ∈ F, the representation
ρ f | Gp is non-split. Moreover the possible exceptions are necessarily non-CM forms.
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6.10.3 Local splitting for Λ-adic forms
Proposition 6.10.5. Let F be a primitive 2-adic Λ-adic form of fixed tame level N
satisfying conditions (1)-(3) above. Then ρ F | Gp splits if and only if F is of CM type.
Proof. The proof is very similar to that for odd primes given in [GV04, Prop. 14]. One
shows that the following statements are equivalent.
1. ρ F | Gp splits.
2. F has infinitely many weight one classical specializations.
3. F has infinitely many weight one classical CM specializations.
4. F is of CM type.
We prove the implications (1) =⇒ (2), to show how the strengthened version of Buzzard’s
result is used. For the remaining implications, we refer to [GV04, Prop. 14]. We remark
that a shortening of the implication (3) =⇒ (4) can be found in [DG10].
(1) =⇒ (2): Recall that we have the following characters:
ψ : Gal( ¯Q/Q) → ¯Q × p
κ : Gal( ¯Q/Q) → Λ × ,
ν : Gal( ¯Q/Q) → Z × p
the character of F of conductor Nq,
the Λ-adic cyclotomic character,
the 2-adic cyclotomic character.
We know that det(ρ F )= ψκν −1 . The specialization of det(ρ F ) at ϕ k,ζ is χν k−1 , where
χ = ψω −k χ ζ . By assumption ρ F | Gp splits, i.e.,
( )
ψκν −1 0
ρ F | Ip ∼
.
0 1
Let P be a weight one point of I extending ϕ 1,ζ : Λ → ¯Q p . It follows that P (ρ F ) = ρ P (F)
has the following shape on I p :
( )
ψω −1 χ
ρ P (F) | Ip ∼
ζ 0
,
0 1
noting that the characters on the diagonal have finite order. Now by Theorem 6.10.2, we
have that
ρ P (F) ∼ ρ f ,
where f is a primitive weight 1 form of level N2 r , with character ψω −1 χ ζ where ζ is
exactly of order 2 r−2 , r ≥ 2. As we vary the point P , and therefore r ≥ 2, we obtain
infinitely many classical weight 1 specializations of F as required.
We remark that elementary arguments (cf. [GV04, (2) =⇒ (3) of Prop. 14]), allow
us to conclude that infinitely many of these must be of CM type, and in particular the
residual representation must necessarily be dihedral! This explains why it is crucial to
assume that Buzzard’s result holds in this case as well.
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6.10.4 Descending to the classical situation
We now give a proof of Theorem 6.10.4.
Proof. If F does not have CM, then by Proposition 6.10.5 the representation ρ F | Gp is
not split. We now show that this forces ρ f | Gp to be non-split for all but except possibly
finitely many arithmetic members f ∈ F. To see this suppose F ∈ I[[q]]. Then
I × = µ d−1 × (1 + m I ),
where d denotes the order of the residue field of I. Since both δ F and ɛ F are I × valued,
we may decompose these characters as δ F = δ t δ w according to the decomposition of I ×
above. Let E t denote the union of the finitely many tamely ramified abelian extensions
of Q p of order dividing d − 1. Let E w denote the maximal abelian pro-p extension of Q p ,
and let E = E t · E w . Set H = Gal( ¯Q p /E). Then δ t and ɛ t (resp., δ w and ɛ w ) are trivial on
Gal( ¯Q p /E t ) (resp., Gal( ¯Q p /E w )). It follows that δ F and ɛ F are trivial on H and hence:
( )
1 λ
ρ F | H ∼ ,
0 1
for some homomorphism λ : H → I.
We show that the homomorphism λ is non-zero. Let c F denote the cocycle corresponding
to λ, as we defined in (6.10.1). Since ρ F | Gp is non-split, the corresponding
cohomology class [c F ] ≠ 0. By [GV04, Lem. 19], the restriction to H of the cohomology
class [c F ] is still non-zero. It follows that the homomorphism λ is non-zero.
Let J denote the non-zero ideal of I generated by the image of λ. Since the intersection
of infinitely many height one primes of I is the zero ideal, we see that J has to be contained
in only finitely many height one primes of I. Let f = P (F) be an arithmetic specialization
of F. Then ρ f | H splits if and only if I ⊂ P . It follows that ρ f | H does not split for all but
finitely many specializations f of F. In particular, ρ f | Gp is non-split for all but finitely
many f of F. This exceptional set does not contain any CM forms by Proposition 6.9.11.
This proves the theorem.
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Appendix A
A 2-adic Control Theorem for
Modular Curves
In this appendix, we study the behaviour of the ordinary parts of the homology groups of
modular curves, associated with the decreasing sequence of congruence subgroups Γ 1 (N2 r )
for r ≥ 2, and we prove a control theorem for these homology groups. As a consequence,
we prove the freeness of W 0 (cf. Theorem 6.5.1 and Theorem A.5.3).
A.1 Introduction
Hida theory studies the modular curves associated to the following congruence subgroups,
for primes p ≥ 5, and (p, N) = 1,
· · · ⊂ Γ 1 (Np r ) ⊂ · · · ⊂ Γ 1 (Np). (∗)
Write Y r for the open complex manifold Γ 1 (Np r )\H. One of the important results in
Hida theory [Hid86a] is that W 0 := lim H ←−r 1 (Y r , Z p ) 0 is a free Λ-module of finite rank and
W 0 /a r W 0 = H 1 (Y r , Z p ) 0 ,
for all r ≥ 1, where a r denotes the augmentation ideal of Z p [[1 + p r Z p ]] and Λ = Z p [[1 +
pZ p ]]. Emerton gives a proof of the results above for p ≥ 5, using algebraic topology
of the Riemann surfaces Y r (cf. [Eme99]). In this appendix, by following the approach
in [Eme99], we prove that the behaviour of the modular curves as in (∗), for p = 2 and 3,
is quite similar to (∗∗) (cf. Theorem A.5.2 and Theorem A.5.3).
The classical versions of the control theorem for p = 2 and 3, do not seem to be
explicitly available in the literature, though an adèlic version of it can be found in [Hid88b].
Emertons proof for primes p ≥ 5 holds for p = 3, with N > 1, verbatim. For the prime
p = 2, a bit more work is needed and forms the content of this appendix. We follow the
notations in the previous chapter (cf. §6.2, §6.3), unless explicitly stated.
(∗∗)
A.2 Notations
Throughout this appendix, let p denote the prime 2 and let q denote 4. Let N be a
natural number coprime to p. We look at the modular curves associated to the following
99

congruence subgroups:
· · · ⊂ Γ 1 (Np r ) ⊂ · · · ⊂ Γ 1 (Nq).
If we take the homology of the tower of modular curves, we get a tower of finitely generated
free abelian groups, which is the abelianization of the above chain of subgroups:
· · · → Γ 1 (Np r ) ab → · · · → Γ 1 (Nq) ab , (A.2.1)
because, for r ≥ 2, H 1 (Y r , Z) = Γ 1 (Np r ) ab . We have a short exact sequence of groups:
1 → Γ 1 (Np r ) → Φ 2 r
η r
→ Γ/Γr → 1,
where η r ( ( a b
c d
)
) = d mod Γr . The action of Φ 2 r on Γ 1 (Np r ) by conjugation induces an
action of Φ 2 r/Γ 1 (Np r ) = Γ/Γ r on Γ 1 (Np r ) ab . Thus Γ acts naturally on Γ 1 (Np r ) ab through
its quotient Γ/Γ r . The morphisms in the chain (A.2.1) are Γ-equivariant.
Remark A.2.1. The restriction of η r to Φ 2 r ∩Γ 0 (p), denoted by Res(η r ), is also surjective
onto Γ/Γ r . Moreover, we have the following commutative diagram:
Φ 2 r+1
η r+1
Γ/Γ r+1
≀
t −1 −t
Φ 2 r ∩ Γ 0 (p) Res(ηr) Γ/Γ r ,
where the group Γ 0 (p) = { ( )
a b
c d ∈ SL2 (Z) | b ≡ 0 (mod 2)}, and t = ( )
1 0
0 p .
For r ≥ s > 1, if we abelianize the short exact sequence in (6.3.1), we obtain:
Γ 1 (Np r ) ab → Φ s ab
r → Γ s /Γ r → 1.
The group Γ s is procyclic, and let γ s denote a generator of it. Then, the augmentation
ideal a s is generated by γ s − 1. For i ≥ 1, the group Γ s+i is generated by γs pi
, and hence
a s+i is generated by γs
pi
− 1. For any r ≥ s, by definition, the augmentation ideal of
Z p [Γ s /Γ r ] is a s . Then for any r ≥ s > 1, by definition
a s Γ 1 (Np r ) ab = [Φ s r, Γ 1 (Np r )]/[Γ 1 (Np r ), Γ 1 (Np r )] ⊂ Γ 1 (Np r ) ab ,
and the last inclusion follows, since Γ 1 (Np r ) is a normal subgroup of Φ s r. The extension
1 → Γ 1 (Np r )/[Φ s r, Γ 1 (Np r )] → Φ s r/[Φ s r, Γ 1 (Np r )] → Γ s /Γ r → 1
is a central extension of a cyclic group, thus the middle group is abelian, implying that
[Φ s r, Φ s r] = [Φ s r, Γ 1 (Np r )]. One way inclusion is clear, because Φ s r ⊇ Γ 1 (Np r ). The other
inclusion follows from the fact that [Φ s r/[Φ s r, Γ 1 (Np r )], Φ s r/[Φ s r, Γ 1 (Np r )]] is trivial.
∼
−→ Γs
Γ r
, by Remark A.2.1. More-
Remark A.2.2. We have an isomorphism
over, we have the following commutative diagram:
Φs r∩Γ 0 (p)
Γ 1 (Np r )∩Γ 0 (p)
Φ s r∩Γ 0 (p) i Φ s
Γ 1 (Np r )∩Γ 0 r
(p) Γ
1 (Np r )
∼
∼
Γ s
Γ r
,
where the map i is induced by the inclusion of the groups Φ s r ∩ Γ 0 (p) ⊆ Φ s r. In fact, the
map i is an isomorphism. This observation is useful in Lemma A.3.5.
100

Summarising this discussion, we see that the map Γ 1 (Np r ) ab → Γ 1 (Np s ) ab in the
chain of homology groups as in (A.2.1) can be factored as the composition of
Γ 1 (Np r ) ab ↠ Γ 1 (Np r ) ab /a s ↩→ Φ s ab
r → Γ 1 (Np s ) ab . (A.2.2)
It may not be true that second and third morphisms of this factorisations are isomorphisms.
But Hida observed that if one applies a certain projection operator arising from
the Atkin U-operator to all these modules then the second and third morphisms become
isomorphisms. So we now define the Atkin U-operator and study their properties.
A.3 Hecke operators
Suppose T is a group which contains subgroups G and H and that t is an element of T
such that t −1 Ht ∩ G has finite index in G. Then one has a transfer map
V : G ab → (t −1 Ht ∩ G) ab .
Conjugation by t induces an isomorphism (t −1 Ht∩G) ab ≃ (H ∩tGt −1 ) ab . Inclusion of the
group H ∩tGt −1 in H induces a morphism (H ∩tGt −1 ) ab → H ab . Taking the composition
of all these we obtain a morphism, we call the ‘Hecke operator’ corresponding to t,
[t] : G ab → H ab .
In the case T = GL 2 (Q), G = H = a congruence subgroup of SL 2 (Z) of level divisible
by p and t = ( ) (
1 0
0 p , we set U := [t]. For A = a b
)
c d ∈ Φ
s
r , we have:
t −1 At = ( a
bp
c/p d
)
and
tAt −1 = ( a b/p
cp d
and hence t −1 Φ s rt ∩ Φ s r = Φ s r ∩ Γ 0 (p), Φ s r ∩ tΦ s rt −1 = Φ s r+1 .
Remark A.3.1. Observe that (1, 1), (2, 2)-entries of A and of t ±1 At ∓1 are the same.
Thus the Atkin U-operator is by definition the composition
)
,
Φ s ab
r
V (Φ s r ∩ Γ 0 (p)) ab t−t −1
∽ Φ s r+1
ab
Φ s ab
r . (A.3.1)
(The final morphism is just that induced by the inclusion of groups Φ s r+1 ⊂ Φs r.) Define
U ′ to be the composition of the first two of above morphisms.
Lemma A.3.2. Suppose that r ≥ s > 1, r ′ ≥ s ′ > 1, r ≥ r ′ , s ≥ s ′ , so that Φ s r ⊂ Φ s′
r ′ .
Then the following diagram commutes:
Φ s ab
r
U ′
Φ s ab
r+1
Φ s′ ab
r ′ U ′
Φ s′ ab
r ′ +1 .
Consequently, the map U commutes with the map Φ s r
ab
is a Z[U]-module via the action of the U-operator.
→ Φ s′ ab
r
. In particular, each Φ s ab
′
r
101

Proof. We can factorize the above diagram as a composition of two diagrams
Φ s ab
r
Φ s′ ab
r ′ V
V
(Φ s ab
r ∩ Γ 0 (p)) ab
(Φ s′ ab
r ′
∩ Γ 0 (p)) ab
t−t −1
Φ s r+1
ab
t−t −1
Φ s′ ab
r ′ +1
The lower portion of this diagram clearly commutes. Since Φ s r ∩ Γ 0 (p) has index p in
Φ s r, we can take { ( )
1 i
0 1 }
p−1
i=0
as the coset representatives. The coset representatives are
independent of the particular values of r and s, and hence the transfer map V is given by
a formula independent of r and s, so the upper portion of the diagram commutes.
A particular case of Lemma A.3.2 is the case r ′ = r − 1, s = s ′ ≤ r ′ . If we write π for
the morphism π : Φ s r
ab −→ Φ s r−1 ab and π′ for the morphism π ′ : Φ s r+1 ab −→ Φs
ab
r , then the
lemma above yields the following formula:
The same definition yields the formula
U ′ ◦ π = π ′ ◦ U ′ = U ∈ End Z (Φ s ab
r ). (A.3.2)
π ◦ U ′ = U ∈ End Z (Φ s ab
r−1).
(A.3.3)
Again by the same lemma, the cokernel of the morphism Γ 1 (Np r ) ab → Φ s ab
r , for r ≥ s > 1,
is a Z[U]-module. Hence Γ s /Γ r is a Z[U]-module.
Lemma A.3.3. For r ≥ s > 1, the operator U acts on Γ s /Γ r as multiplication by p.
Proof. The operator U acts on Γ s /Γ r as a multiplication by p if and only if it acts on
Φ s ab
r
Γ 1 (Np r ) ab
Φ s ab
r
Γ 1 (Np r ) ab
by Ā ↦→ Āp . The operator U is the composition of the following maps:
V
−→
(Φ s r ∩Γ0 (p)) ab t−t
(Γ 1
−→
−1
(Np r )∩Γ 0 (p)) ab
Φ s ab
r+1
Φ r ab
r+1
−→
Φ s ab
r
Γ 1 (Np r ) ab
Ā ↦−→ Ā p ↦−→ tĀp t −1 ↦−→ tĀp t −1 . (A.3.4)
Let {α i = ( )
1 i
0 1 }
p−1
i=0 be a set of coset representatives of the group Φs r ∩ Γ 0 (p) in Φ s r. With
this choice, we see that the transfer map in (A.3.4) looks like Ā ↦→ Āp . By Remark A.3.1,
tA p t −1 and A p represent the same coset mod Γ 1 (Np r ) ab , and hence we are done.
We would like to define an action of Γ on Φ s r
ab and call it the nebentypus action.
For r ≥ 2, if ¯d ∈ Γ/Γ r , then choose an element α ∈ Φ 2 r+1 ∩ Γ0 (p), such that η r (α) = ¯d
(cf. Remark A.2.1). The nebentypus action of d on Φ s r
ab is given by conjugation by
α. This action is well-defined because, if α 1 and α 2 denote two lifts of ¯d, then α1 −1 α 2 ∈
Γ 1 (Np r+1 ) ∩ Γ 0 (p) ⊆ Φ s r, and hence for any element x ∈ Φ s r, α1 −1 α 2xα2 −1 α 1 = x in Φ s r ab .
Now, we shall show that the actions of Γ and U commutes.
Lemma A.3.4. If r ≥ s > 1, the action of U commutes with the action of Γ on Φ s ab
r .
102

Proof. It is easy to see that α(Φ s r ∩Γ 0 (p))α −1 = Φ s r ∩Γ 0 (p) for any α ∈ Φ 2 r+1 ∩Γ0 (p), since
αΦ s rα −1 ⊆ Φ s r. We need to prove that the diagram below is commutative, because in the
diagram below composition of the vertical morphisms on either side are the operator U
and it commutes with the automorphism of Φ s r
ab induced by conjugation by α, but we
know that Γ acts on Φ s r ab by conjugation by such elements α.
Φ s ab
r
α−α −1
Φ s ab
r
V
(Φ s r ∩ Γ 0 (p))
ab
α−α−1
(α(Φ s r ∩ Γ 0 (p))α −1 ) ab = (Φ s r ∩ Γ 0 (p)) ab
V
Φ s ab
r+1
t−t −1
α−α −1
αtα −1 (−)αt −1 α −1
(αΦ s r+1 α−1 ) ab = Φ s ab
r+1
Φ s ab
r
α−α −1
(αΦ s rα −1 ) ab = Φ s ab
The top square in the diagram above commutes because, if {γ 1 , . . . , γ p } form a set of coset
representatives for the group Φ s r ∩ Γ 0 (p) in Φ s r, then so is {αγ 1 α −1 , . . . , αγ p α −1 }. This
diagram commutes even if α ∈ Φ 2 r ∩ Γ 0 (p). The last square commutes by the functoriality
of the transfer map. We now prove the commutativity of the middle square, i.e., the map
αtα −1 (−)αt −1 α −1 : (Φ s r∩Γ 0 (p)) ab → Φ s r+1 ab is t−t−1 . For g ∈ Φ s r∩Γ 0 (p), αtα −1 gαt −1 α −1 =
(αtα −1 t −1 )tgt −1 (αtα −1 t −1 ) −1 . For α ∈ Φ 2 r+1 ∩ Γ0 (p), we have αtα −1 t −1 ∈ Γ 1 (Np r+1 ) ⊆
Φ s r+1 . Thus conjugation by αtα−1 t −1 induces the identity on Φ s r+1 ab.
Lemma A.3.5. The transfer morphism V : Φ s r
ab
of U on its source and target.
Φ s ab
r
r .
→ Γ 1 (Np r ) ab commutes with the action
Proof. The inclusion of groups Γ 1 (Np r ) ⊆ Φ s r gives rise to the another transfer map
V
→ Γ 1 (Np r ) ab . It suffices to prove that the following diagram commutes:
Φ s ab
r
V
Φ r ab
r
V
(Φ s r ∩ Γ 0 (p)) ab V (Γ 1 (Np r ) ∩ Γ 0 (p)) ab
Φ s ab
r
t−t −1
V
t−t −1
V Γ 1 (Np r ) ab .
Here V denotes the transfer maps between various abelianizations and to be understood
from the context. The commutativity of the top square and the bottom square follows
from the functoriality of the transfer map, and from Remark A.2.2, Remark A.3.1, respectively.
If {σ d } denote a set of coset representatives of Γ 1 (Np r ) ∩ Γ 0 (p) in Φ s r ∩ Γ 0 (p),
then the conjugates {tσ d t −1 } form a set of coset representatives of Γ 1 (Np r ) in Φ s r.
To summarise, we have defined the U-operators for the congruence subgroups {Φ s r}
and we have checked that morphisms between these congruence subgroups respects the
action of U and this action commutes with that of Γ. We have also checked that the
transfer morphism V : Φ s ab
r → Γ 1 (Np r ) ab commutes with the action of U.
103

A.4 Ordinary parts
Let M be a Z p [X]-module, which is finitely generated as a Z p -module. Then, we have a
morphism of Z p -modules
Z p [X] → End Zp (M).
The endomorphism ring End Zp (M) of M is a finite Z p -algebra. Thus the image of Z p [X]
in End Zp (M), will be denoted by A, is also a commutative finite Z p -algebra, and hence it
factors as a product of local rings. Let A 0 denote the product of all those local factors of A
in which the image of X is a unit, and A n its complementary factor, so that A = A 0 × A n .
Each of these is a flat A-algebra, and a subalgebra of End Zp (M). We define
M 0 := M ⊗ A A 0
and call this the ordinary part of M. Clearly, taking ordinary parts is an exact functor.
If we consider X to be the U-operator, we may consider the ordinary part of the
Z p -homology of the curve Y r , i.e., the module (Γ 1 (Np r ) ab ⊗ Z p ) 0 , which is a Γ-module by
Lemma A.3.4. We have the following fundamental theorem for the prime p = 2 and it is
similar to the theorem, which was proved in [Hid86a] for primes p ≥ 5.
Theorem A.4.1. If r ≥ s > 1, then the morphism of abelian groups
is an isomorphism.
(Γ 1 (Np r ) ⊗ Z p ) 0 /a s ↠ (Γ 1 (Np s ) ⊗ Z p ) 0
Proof. We shall prove the following isomorphisms:
(Γ 1 (Np r ) ab ⊗ Z p ) 0 /a s
∼ −→ (Φ
s ab
r ⊗ Z p ) 0 ∼ −→ (Γ1 (Np s ) ab ⊗ Z p ) 0 .
We have constructed an operator U ′ : Φ s ab
r−1
and (A.3.3). If π : Φ s ab
r
Φ s ab
r
→ Φ s ab
r−1
, satisfying the equations (A.3.2)
is the morphism induced by the inclusion of groups
→ Φs
ab
r
⊂ Φ s r−1 ab,
then U ′ ◦ π = U ∈ End(Φ s r
ab ), π ◦ U ′ = U ∈ End(Φ s r−1).
ab
We have the following diagram:
Φ s r−1
ab π
Φ s r−2
ab
U ′ U
U ′
Φ s r
ab π
Φ s r−1
ab
U ′ U
U ′
Φ s ab
r+1
π
Φ s r ab .
The existence of U ′ implies that upon tensoring over Z p and taking the ordinary parts π
induces an isomorphism (Φ s r
ab ) 0 = (Φ s r−1 ab)0
. By descending induction on r, we obtain the
required isomorphism
(Φ s r
ab ) 0 = (Φ s s ab ) 0 = (Γ 1 (Np s ) ab ⊗ Z p ) 0 .
104

To prove the first isomorphism, we consider the short exact sequence:
1 → Γ 1 (Np r ) ab /a s → Φ s ab
r → (Γ s /Γ r ) → 1.
Tensor this short exact sequence with Z p and then take the ordinary parts to obtain:
1 → (Γ 1 (Np r ) ab ⊗ Z p ) 0 /a s → (Φ s ab
r ⊗ Z p ) 0 → (Γ s /Γ r ) 0 → 1,
because Z p is flat over Z and taking ordinary parts preserves exactness. By Lemma A.3.3,
the operator U is nilpotent, because Γ s /Γ r is a p-torsion group. Thus Γ s /Γ r has trivial
ordinary part and hence the Theorem follows.
A.5 Iwasawa modules
We have the following inverse system indexed by natural numbers r ≥ 2,
Define the Iwasawa module by
· · · → Γ 1 (Np r ) ab ⊗ Z p → · · · → Γ 1 (Nq) ab ⊗ Z p .
W := lim ←− r≥2
Γ 1 (Np r ) ab ⊗ Z p .
The profinite group Γ acts on the Z p -module Γ 1 (Np r ) ab ⊗ Z p through its finite quotient
Γ/Γ r . Thus the Iwasawa module W becomes a module over the completed group algebra
Λ := Z p [[Γ]] = lim Z ←−r≥2 p [Γ/Γ r ]. The Iwasawa module W is difficult to understand, because
we do not have a good characterisation of the image of the morphism
Γ 1 (Np r ) ab → Γ 1 (Np s ) ab
(r ≥ s > 1) in general, and so we cannot get a good description of the projective limit.
However, Theorem A.4.1 allows us to understand the ordinary part of W very well. To
make the statement clear, let us slightly abstract the situation.
Let {M r } r≥2 be a system of Λ-modules, such that each M r is pointwise fixed by Γ r ,
i.e., M r is a module over Λ/a r Λ = Z p [Γ/Γ r ]. For each r ≥ s > 1, we have a map M r → M s
such that it factors via
M r /a s M r → M s .
Define W := lim ←−r≥2
M r . For each r ≥ 2, we have a map W → M r , and they factor as
W/a r W → M r .
Lemma A.5.1. Assume that each M r is p-adically complete and for each r ≥ s > 1,
M r /a s M r → M s is an isomorphism. Then W/a s W ↠ M s is an isomorphism.
Proof. Clearly the maps M r → M s are surjective for all r ≥ s, and hence the canonical
map from W → M s is also surjective. It is enough to show that the kernel is a s W . Since
each M r is p-adically complete and we have that M r = lim M ←−i r /p i M r . Moreover, we have
that each M r is pointwise fixed by Γ r and hence M r = lim M ←−i r /n i M r , where γ r is a cyclic
generator of Γ r and n = (γ r − 1, p).
105

By induction on i, we see that γs
pi
− 1/γ s − 1 is an element of the ideal (γ s − 1, p) i . In
particular, when s = 2, the element γ pr−2
2 − 1/γ 2 − 1 is an element of the maximal ideal
m = (γ 2 − 1, p) and hence n ⊆ m = (a 2 , p). Since m pr−2 ⊆ ((γ 2 − 1) pr−2 , p) ⊆ n ⊆ m,
we see that each M r is m-adically complete, since they are n-adically complete. Once we
have that each M r is m-adically complete, then proving the injectivity of the map above
is quite similar to the proof of [Eme99, Prop. 5.1].
The following Theorem is an immediate consequence of the above Lemma.
Theorem A.5.2. For any r ≥ 2, we have
is the Γ r -coinvariants of W 0 .
W 0 /a r W 0 ∼ = (Γ1 (Np r ) ab ⊗ Z p ) 0
Proof. This follows from Lemma A.5.1 together with Theorem A.4.1
The theorem above is one of the key step in the proof of the main theorem A.5.3.
Each module Γ 1 (Np r ) ab ⊗ Z p is free of finite rank over Z p , and so is compact in its p-adic
topology. Thus if we give the limiting module W the topology which is the projective
limit of the p-adic topologies on each module Γ 1 (Np r ) ab ⊗ Z p it becomes a compact Λ-
module. Furthermore, Λ acts continuously on W , since Γ acts on each of the modules
Γ 1 (Np r ) ab ⊗ Z p through a finite quotient. Since W 0 is a direct factor of W the same
remarks hold true for W 0 . Furthermore, Theorem A.5.2 implies that the projective limit
topology on W 0 coincides with its m-adic topology (where m = (a 2 , p) ⊂ Λ denotes the
maximal ideal of Λ), because the kernels of the projection Λ → Z p /p r Z p [Γ/Γ r ] are cofinal
with the sequence of ideals m r in Λ. Thus W 0 is a Λ-module, compact in its m-adic
topology such that
W 0 /m = W 0 /(a 2 , p) = (Γ 1 (Nq) ab ⊗ Z p /p) 0
is a finite dimensional Z p /pZ p -module, of dimension d (say). By Nakayama’s lemma, we
have that W 0 is a finitely generated Λ-module with a minimal generating set of order
equal to d. By Theorem A.5.3 below, d will be equal to the Z p -rank of the free Z p -module
(Γ 1 (Np r ) ab ⊗ Z p ) 0 , for r ≥ 2. We have the following theorem for the prime p = 2 and it
is similar to the theorem, which was proved in [Hid86a] for primes p ≥ 5.
Theorem A.5.3. The Λ-module W 0 is free of finite rank equal to d.
By [NSW08, Prop. 5.1.9], it is enough to show that, the module W 0 is a reflexive Λ-
module. We will prove this by considering the duality theory of the modules (Γ 1 (Np r ) ab ⊗
Z p ) 0 and showing that they are reflexive as Z p [Γ/Γ r ]-modules.
We briefly recall some results from Duality theory. Suppose R is a commutative ring,
G is a finite group, and M is a left R[G]-module. Write M ∗ = Hom R (M, R), the R-dual
of M. The module M ∗∗ can be made a left module over R[G]. By definition of M ∗ ,
there is a natural morphism of left R[G]-modules M → M ∗∗ . The module M is called
reflexive, if this map is an isomorphism. The following lemma will be useful later.
Lemma A.5.4 ([Eme99], Lem. 6.2). If M is reflexive as an R-module, then M is reflexive
as an R[G]-module.
Now, we need to understand how to use the reflexivity results for modules over
Z p [Γ/Γ r ] to show the reflexivity of W 0 as a Λ-module.
106

A.6 Limits of cohomology modules
Cohomology is the dual of homology:
H 1 (Y r , Z p ) := Hom Z (Γ 1 (Np r ) ab , Z p ) = Hom Zp (Γ 1 (Np r ) ab ⊗ Z p , Z p ).
The ring Λ acts on Γ 1 (Np r ) ab ⊗ Z p through its quotient Λ r := Λ/a r = Z p [Γ/Γ r ]. If
r ≥ s > 1 then the ring Λ s is the quotient of the ring Λ r by the augmentation ideal a s ,
i.e., Λ r ↠ Λ s = Λ r /a s . Thus we get the following sequence of morphisms of Λ r -modules:
Hom Λr (Γ 1 (Np r ) ab ⊗ Z p , Λ r ) → Hom Λr (Γ 1 (Np r ) ab ⊗ Z p , Λ r )/a s
→ Hom Λr (Γ 1 (Np r ) ab ⊗ Z p , Λ s ) = Hom Λs (Γ 1 (Np r ) ab ⊗ Z p /a s , Λ s ).
If M is any Z p [U]-module, which is finitely generated as a Z p -module, then so is the
Z p -dual M ∗ := Hom Zp (M, Z p ) (via the dual action of U). Clearly (M ∗ ) 0 = (M 0 ) ∗ , i.e.,
taking ordinary parts commutes with taking duals. By Theorem A.4.1 and by taking the
ordinary parts of the homomorphisms above, we obtain a morphism:
Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r )/a s
Lemma A.6.1. The morphism η is an isomorphism.
η
−→ Hom Λs ((Γ 1 (Np s ) ab ⊗ Z p ) 0 , Λ s ).
Proof. By virtue of Lemma A.3.5, we may restrict V to the ordinary parts of its source
and target to obtain a morphism which we continue to denote by V
(Φ s ab
r ⊗ Z p ) 0 V
−→ (Γ 1 (Np r ) ab ⊗ Z p ) 0 .
There is a dual morphism Hom Zp ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Z p ) −→ V ∗
Hom Zp ((Φ s r ab ⊗ Z p ) 0 , Z p ),
which sits in the first column of the following commutative diagram:
Hom Zp ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Z p )
V ∗
Hom Zp ((Φ s r ab ⊗ Z p ) 0 , Z p )
∼
Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r )
Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r )/a s
≀
Hom Λs ((Γ 1 (Np r ) ab ⊗ Z p ) 0 /a s , Λ s )
Hom Zp ((Γ 1 (Np s ) ab ⊗ Z p ) 0 , Z p )
∼ Hom Λs ((Γ 1 (Np s ) ab ⊗ Z p ) 0 , Λ s )
≀
in which the two horizontal isomorphisms are those provided by Lemma A.5.4, and the
two vertical isomorphisms follow from Theorem A.4.1 and its proof. Thus to prove the
lemma it suffices to prove that
Hom Zp ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Z p ) V ∗
−→ Hom Zp ((Φ s ab
r ⊗ Z p ) 0 , Z p ) (A.6.1)
is surjective with kernel equal to a s (Hom Zp ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Z p )). Since V commutes
with U and taking ordinary parts commutes with taking Z p -duals, we see that the morphism
in (A.6.1) is the ordinary part of the morphism
Hom Zp ((Γ 1 (Np r ) ab ⊗ Z p ), Z p ) V ∗
−→ Hom Zp ((Φ s ab
r ⊗ Z p ), Z p ).
107

Taking ordinary parts is also exact and commutes with the action of Γ. Thus it suffices to
show that the map above is surjective with kernel equal to a s Hom Zp ((Γ 1 (Np r ) ab ⊗Z p ), Z p ).
This statement was proved in [Eme99, §8] for torsion-free groups H and G such that
H ⊆ G, instead of Γ 1 (Np r ) ⊆ Φ s r. It is known that the group Γ 1 (M) is torsion free for
all M ≥ 4. In particular, the group Γ 1 (Nq) is torsion-free, and so is the group Φ s r.
We now give a proof of Theorem A.5.3.
Lemma A.6.2. There is a canonical isomorphism
Hom Λ (W 0 , Λ) = lim ←− r≥2
Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r ).
Proof. We have the following series of canonical isomorphisms:
Hom Λ (W 0 , Λ) = lim Hom ←−r Λ (W 0 , Λ r ) = lim Hom ←−r Λr (W 0 /a r , Λ r )
= lim ←−r
Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r ),
where the last isomorphism follows from the Theorem A.5.2.
Proposition A.6.3. For r > 1, there is a canonical isomorphism
Hom Λ (W 0 , Λ)/a r = Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r ).
Proof. This follows from Lemma A.6.1, Lemma A.6.2 and Lemma A.5.1.
Theorem A.6.4. The module W 0 is Λ-free.
Proof. Since any finitely generated reflexive Λ-module is free, it suffices to show that W 0
is a reflexive Λ-module. This follows from the following series of canonical isomorphisms:
Hom Λ (Hom Λ (W 0 , Λ), Λ) = lim ←−r
Hom Λ (Hom Λ (W 0 , Λ), Λ r )
= lim ←−r
Hom Λ (Hom Λr (W 0 , Λ)/a r , Λ r )
(1)
= lim ←−r
Hom Λr (Hom Λr ((Γ 1 (Np r ) ab ⊗ Z p ) 0 , Λ r ), Λ r )
(2)
= lim ←−r
(Γ 1 (Np r ) ab ⊗ Z p ) 0 = W 0 .
(Equality (1) follows from Proposition A.6.3. Equality (2) follows from Lemma A.5.4,
since (Γ 1 (Np r ) ab ⊗Z p ) 0 is a free Z p -module and so is certainly a reflexive Z p -module).
108

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