Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 3
is assumed to be isomorphic to a subgroup of GL 2 (F l )thatcontains SL 2 (F l ), E F
is assumed to have stable reduction at all of the nonarchimedean valuations of F ,
def
C K = C F × F K is assumed to be a K-core [cf. [CanLift], Remark 2.1.1], V
is assumed to be a collection of valuations of K such that the natural inclusion
F mod ⊆ F ⊆ K induces a bijection V → ∼ V mod between V and the set V mod of all
valuations of the number field F mod ,and
V bad
mod
⊆ V mod
is assumed to be some nonempty set of nonarchimedean valuations of odd residue
characteristic over which E F has bad [i.e., multiplicative] reduction — i.e., roughly
speaking, the subset of the set of valuations where E F has bad multiplicative reduction
that will be “of interest” to us in the context of the theory of the present series
of papers. Then we shall write V bad def
= V bad
mod × V mod
V ⊆ V, V good def
mod
= V mod \ V bad
mod ,
V good def
= V\V bad . Also, we shall apply the superscripts “non” and “arc” to V, V mod
to denote the subsets of nonarchimedean and archimedean valuations, respectively.
This data determines, up to K-isomorphism [cf. Remark 3.1.3], a finite étale
covering C K → C K of degree l such that the base-changed covering
X K
def
def
= C K × CF X F → X K = X F × F K
arises from a rank one quotient E K [l] ↠ Q ( ∼ = Z/lZ) of the module E K [l]ofl-torsion
points of E K (K) which,atv ∈ V bad , restricts to the quotient arising from coverings
of the dual graph of the special fiber. Moreover, the above data also determines a
cusp
ɛ
of C K which, at v ∈ V bad , corresponds to the canonical generator, upto±1, of Q
[i.e., the generator determined by the unique loop of the dual graph of the special
fiber]. Furthermore, at v ∈ V bad , one obtains a natural finite étale covering of
degree l
X v
→ X v
def
= X K × K K v (→ C v
def
= C K × K K v )
by extracting l-th roots of the theta function; at v ∈ V good , one obtains a natural
finite étale covering of degree l
X−→ v
→ X v
def
= X K × K K v (→ C v
def
= C K × K K v )
determined by ɛ. More details on the structure of the coverings C K , X K , X v
[for
v ∈ V bad ], −→v X [for v ∈ V good ] may be found in [EtTh], §2, as well as in §1 ofthe
present paper.
In this situation, the objects
l
def
= (l − 1)/2; l ± def
= (l +1)/2; F l
def
= F × l
/{±1}; F⋊±
l
def
= F l ⋊ {±1}