Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 35
Section 1: Complements on Coverings of Punctured Elliptic Curves
In the present §1, we discuss certain routine complements — which will be of
use in the present series of papers — to the theory of coverings of once-punctured
elliptic curves, as developed in [EtTh], §2.
Let l ≥ 5beaninteger prime to 6; X a hyperbolic curve of type (1, 1) over a
field k of characteric zero; C a hyperbolic orbicurve of type (1,l-tors) ± [cf. [EtTh],
Definition 2.1] over k, whosek-core [cf. [CanLift], Remark 2.1.1; [EtTh], the discussion
at the beginning of §2] also forms a k-core of X. Thus,C determines, up to
k-isomorphism, a hyperbolic orbicurve X of type (1,l-tors) [cf. [EtTh], Definition
2.1] over k. Moreover, if we write G k for the absolute Galois group of k [relative to
an appropriate choice of basepoint], Π (−) for the arithmetic fundamental group of a
geometrically connected, geometrically normal, generically scheme-like k-algebraic
stack of finite type “(−)” [i.e., the étale fundamental group π 1 ((−))], and Δ (−) for
the geometric fundamental group of “(−)” [i.e., the kernel of the natural surjection
Π (−) ↠ G k ], then we obtain natural cartesian diagrams
X −→ X
⏐ ⏐
↓ ↓
C −→ C
Π X −→ Π X Δ X −→ Δ X
⏐ ⏐ ⏐ ⏐
↓ ↓ ↓ ↓
Π C −→ Π C Δ C −→ Δ C
of finite étale coverings of hyperbolic orbicurves and open immersions of profinite
groups. Finally, let us make the following assumption:
(∗) The natural action of G k on Δ ab
X ⊗ (Z/lZ) [where the superscript “ab”
denotes the abelianization] is trivial.
Next, let ɛ be a nonzero cusp of C — i.e., a cusp that arises from a nonzero
element of the quotient “Q” that appears in the definition of a “hyperbolic orbicurve
of type (1,l-tors) ± ” given in [EtTh], Definition 2.1. Write ɛ 0 for the unique “zero
cusp” [i.e., “non-nonzero cusp”] of X; ɛ ′ , ɛ ′′ for the two cusps of X that lie over ɛ;
and
Δ X ↠ Δ ab
X ⊗ (Z/lZ) ↠ Δ ɛ
for the quotient of Δ ab
X ⊗ (Z/lZ) by the images of the inertia groups of all nonzero
cusps ≠ ɛ ′ ,ɛ ′′ of X. Thus, we obtain a natural exact sequence
0 −→ I ɛ ′ × I ɛ ′′ −→ Δ ɛ −→ Δ E ⊗ (Z/lZ) −→ 0
—wherewewriteE for the genus one compactification of X, and I ɛ ′, I ɛ
′′ for
the respective images in Δ ɛ of the inertia groups of the cusps ɛ ′ , ɛ ′′ [so we have
noncanonical isomorphisms I ɛ ′
∼ = Z/lZ ∼ = Iɛ ′′].
Next, let us observe that G k , Gal(X/C) ( ∼ = Z/2Z) act naturally on the above
exact sequence. Write ι ∈ Gal(X/C) for the unique nontrivial element. Then ι
induces an isomorphism I ɛ ′
∼ = Iɛ ′′; if we use this isomorphism to identify I ɛ ′, I ɛ ′′,
then one verifies immediately that ι acts on the term “I ɛ
′ × I ɛ
′′” oftheaboveexact
sequence by switching the two factors. Moreover, one verifies immediately that ι