Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 53
the construction of the coverings “X”, “C” in the discussion at the beginning
of [EtTh], §2 [cf. also [AbsAnab], Lemma 1.1.4, (i)], that the closed subgroup
Δ X ⊆ Π CK may be characterized by a rather simple explicit algorithm. Since
the decomposition groups of Π CK at the nonzero cusps — i.e., the cusps whose
inertia groups are contained in Δ X [cf. the discussion at the beginning of §1] —
are also group-theoretic [cf., e.g., [AbsTopI], Lemma 4.5], the above observation
follows immediately from the easily verified fact that the image of any of these
decomposition groups associated to nonzero cusps coincides with the image of Π XK
in Π CK /Δ X .
(ii) In light of the observation of (i), it makes sense to adopt the following
convention:
Instead of applying the group-theoretic reconstruction algorithm of [AbsTopIII],
Theorem 1.9 [cf. also the discussion of [AbsTopIII], Remark
2.8.3], directly to Π CK [or topological groups isomorphic to Π CK ], we
shall apply this reconstruction algorithm to the open subgroup Π XK ⊆
Π CK to reconstruct the function field of X K , equipped with its natural
Gal(X K /C K ) ∼ = Π CK /Π XK -action.
In this context, we shall refer to this approach of applying [AbsTopIII], Theorem 1.9,
as the Θ-approach to [AbsTopIII], Theorem 1.9. Note that, for v ∈ V good ⋂ V non
(respectively, v ∈ V bad ), one may also adopt a “Θ-approach” to applying [AbsTopIII],
Theorem 1.9, to Π Cv or [by applying Corollary 1.2] Π X−→v ,Π C−→v (respectively,
to Π tp
C
or [by applying [EtTh], Proposition 2.4] Π tp
v
X
). In the present series
v
of papers, we shall always think of [AbsTopIII], Theorem 1.9 [as well as the other
results of [AbsTopIII] that arise as consequences of [AbsTopIII], Theorem 1.9] as being
applied to [isomorphs of] Π CK or, for v ∈ V good ⋂ V non (respectively, v ∈ V bad ),
Π Cv ,Π X−→v ,Π C−→v (respectively, Π tp
C
,Π tp
v
X
) via the “Θ-approach”.
v
(iii) Recall from the discussion at the beginning of [EtTh], §2, the tautological
extension
1 → Δ Θ → Δ Θ X → Δ ell
X → 1
def
—whereΔ Θ = [Δ X , Δ X ]/[Δ X , [Δ X , Δ X ]]; Δ Θ def
X = Δ X /[Δ X , [Δ X , Δ X ]]; Δ ell def
X =
Δ ab
X . The extension class ∈ H2 (Δ ell
X , Δ Θ) of this extension determines a tautological
isomorphism
M X
∼
→ ΔΘ
— where we recall from [AbsTopIII], Theorem 1.9, (b), that the module “M X ”of
[AbsTopIII], Theorem 1.9, (b) [cf. also [AbsTopIII], Proposition 1.4, (ii)], may be
naturally identified with Hom(H 2 (Δ ell
X , Ẑ), Ẑ). In particular, we obtain a tautological
isomorphism
∼
M X → (l · ΔΘ )
[i.e., since [Δ X :Δ X ]=l]. From the point of view of the theory of the present series
of papers, the significance of the “Θ-approach” lies precisely in the existence of this
∼
tautological isomorphism M X → (l · ΔΘ ), which will be applied in [IUTchII] at v ∈