Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

54 SHINICHI MOCHIZUKI
V bad . That is to say, the Θ-approach involves applying the reconstruction algorithm
of [AbsTopIII], Theorem 1.9, via the cyclotome M X , which may be identified, via
the above tautological isomorphism, with the cyclotome (l · Δ Θ ), which plays a
central role in the theory of [EtTh] — cf., especially, the discussion of “cyclotomic
rigidity” in [EtTh], Corollary 2.19, (i).
(iv) If one thinks of the prime number l as being “large”, then the role played
by the covering X in the above discussion of the “Θ-approach” is reminiscent of the
role played by the universal covering of a complex elliptic curve by the complex plane
in the holomorphic reconstruction theory of [AbsTopIII], §2 [cf., e.g., [AbsTopIII],
Propositions 2.5, 2.6].
Remark 3.1.3. Since V bad
mod ≠ ∅ [cf. Definition 3.1, (b)], it follows immediately
from Definition 3.1, (d), (e), (f), that the data (F/F, X F ,l,C K , V, V bad
mod ,ɛ)is,
in fact, completely determined by the data (F/F, X F , C K , V, V bad
mod
), and that
C K is completely determined up to K-isomorphism by the data (F/F, X F , l, V).
Finally, we remark that for given data (X F , l, V bad
mod
), distinct choices of “V” will
not affect the theory in any significant way.
Remark 3.1.4. It follows immediately from the definitions that at each v ∈ V bad
[which is necessarily prime to l — cf. Definition 3.1, (c)] (respectively, each v ∈
V good ⋂ V non which is prime to l; each v ∈ V good ⋂ V non ), X v
(respectively, −→v X ;
X v )admitsastable model over the ring of integers of K v .
Remark 3.1.5. Notethatsincethe3-torsion points of E F are rational over F ,
and F is Galois over F mod [cf. Definition 3.1, (b)], it follows [cf., e.g., [IUTchIV],
Proposition 1.8, (iv)] that K is Galois over F mod . In addition to working with
the field F mod and various extensions of F mod contained in F , we shall also have
occasion to work with the algebraic stack
S mod
def
= Spec(O K ) // Gal(K/F mod )
obtained by forming the stack-theoretic quotient [i.e., “//”] of the spectrum of the
ring of integers O K of K by the Galois group Gal(K/F mod ). Thus, any finite extension
L ⊆ F of F mod in F determines, by forming the integral closure of S mod in L,
an algebraic stack S mod,L over S mod . In particular, by considering arithmetic line
bundles over such S mod,L , one may associate to any quotient Gal(F/F mod ) ↠ Q
a Frobenioid via [the easily verified “stack-theoretic version” of] the construction
of [FrdI], Example 6.3. One verifies immediately that an appropriate analogue of
[FrdI], Theorem 6.4, holds for such stack-theoretic versions of the Frobenioids constructed
in [FrdI], Example 6.3. Also, we observe that upon passing to either the
perfection or the realification, such stack-theoretic versions become naturally isomorphic
to the non-stack-theoretic versions [i.e., of [FrdI], Example 6.3, as stated].
Remark 3.1.6. In light of the important role played by the various orbicurves
constructed in [EtTh], §2, in the present series of papers, we take the opportunity
to correct an unfortunate — albeit in fact irrelevant! — error in [EtTh]. In the