Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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34 SHINICHI MOCHIZUKI Suppose that for i =1, 2, C i and C i ′ are categories. Then we shall say that two isomorphism classes of functors φ : C 1 →C 2 , φ ′ : C 1 ′ →C 2 ′ are abstractly equivalent ∼ if, for i =1, 2, there exist isomorphisms α i : C i →C ′ i such that φ ′ ◦ α 1 = α 2 ◦ φ. We shall also apply this terminology to morphisms between temperoids, as well as to morphisms between subcategories of connected objects of temperoids. Numbers: We shall use the abbreviations NF (“number field”), MLF (“mixed-characteristic [nonarchimedean] local field”), CAF (“complex archimedean field”), RAF (“real archimedean field”), AF (“archimedean field”) as defined in [AbsTopI], §0; [AbsTopIII], §0. We shall denote the set of prime numbers by Primes. Let F be a number field [i.e., a finite extension of the field of rational numbers]. Then we shall write V(F ) = V(F ) ⋃ arc V(F ) non for the set of valuations of F , i.e., the union of the sets of archimedean [i.e., V(F ) arc ] and nonarchimedean valuation [i.e., V(F ) non ]ofF . Let v ∈ V(F ). Then we shall write F v for the completion of F at v; if, moreover, L is any [possibly infinite] Galois extension of F , then, by a slight abuse of notation, we shall write L v for the completion of L at some valuation ∈ V(L) that lies over v. If v ∈ V(F ) non ,then we shall write p v for the residue characteristic of v. If v ∈ V(F ) arc , then we shall write p v ∈ F v for the unique positive real element of F v whose natural logarithm is equal to 1 [i.e., “e =2.71828 ...”]. By passing to appropriate projective or inductive limits, we shall also apply the notation “V(F )”, “F v ”, “p v ” in situations where “F ” is an infinite extension of Q. Curves: We shall use the terms hyperbolic curve, cusp, stable log curve, andsmooth log curve as they are defined in [SemiAnbd], §0. We shall use the term hyperbolic orbicurve as it is defined in [Cusp], §0.
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
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(v) Write INTER-UNIVERSAL TEICHMÜL
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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