Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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14 SHINICHI MOCHIZUKI — in essence, a system of Frobenioids — associated to this initial Θ-data, as well as an associated D-Θ ±ell NF-Hodge theater † HT D-Θ±ell NF — in essence, the system of base categories associated to the system of Frobenioids † HT Θ±ell NF . (i) (F ⋊± l - and F l -Symmetries) The Θ±ell NF-Hodge theater † HT Θ±ell NF may be obtained as the result of gluing together a Θ ±ell -Hodge theater † HT Θ±ell to a ΘNF-Hodge theater † HT ΘNF [cf. Remark 6.12.2, (ii)]; a similar statement holds for the D-Θ ±ell NF-Hodge theater † HT D-Θ±ellNF .Theglobal portion of a D-Θ ±ell - Hodge theater † HT D-Θ±ell consists of a category equivalent to [the full subcategory determined by the connected objects of] the Galois category of finite étale coverings of the [orbi]curve X K . This global portion is equipped with a F ⋊± l -symmetry, i.e., a poly-action by F ⋊± l on the labels ( −l < ... < −1 < 0 < 1 < ... < l ) — which we think of as elements ∈ F l — each of which is represented in the D- Θ ±ell -Hodge theater † HT D-Θ±ell by a D-prime-strip [cf. Fig. I1.3]. The global portion of a D-ΘNF-Hodge theater † HT D-ΘNF consists of a category equivalent to [the full subcategory determined by the connected objects of] the Galois category of finite étale coverings of the orbicurve C K . This global portion is equipped with a F l -symmetry, i.e., a poly-action by F l on the labels (1 < ... < l ) — which we think of as elements ∈ F l — each of which is represented in the D-ΘNF-Hodge theater † HT D-ΘNF by a D-prime-strip [cf. Fig. I1.3]. The D- Θ ±ell -Hodge theater † HT D-Θ±ell is glued to the D-ΘNF-Hodge theater † HT D-ΘNF along a single D-prime-strip in such a way that the labels 0 ≠ ±t ∈ F l that arise in the F ⋊± l -symmetry are identified with the corresponding label j ∈ F l that arises in the F l -symmetry. (ii) (Θ-links) By considering the 2l-th roots of the q-parameters “q v ”of the elliptic curve E F at v ∈ V bad and extending to other v ∈ V in such a way as to satisfy the product formula, one may construct a natural F ⊩ -prime-strip † F ⊩ mod associated to the Θ±ell NF-Hodge theater † HT Θ±ellNF . In a similar vein, by considering the reciprocal of the l-th root of the Frobenioid-theoretic theta function “Θ v ” associated to the elliptic curve E F at v ∈ V bad and extending to other v ∈ V in such a way as to satisfy the product formula, one may construct a natural F ⊩ -prime-strip † F ⊩ tht associated to the Θ±ell NF-Hodge theater † HT Θ±ellNF .Now let ‡ HT Θ±ellNF be another Θ ±ell NF-Hodge theater [relative to the given initial Θ- data]. Then we shall refer to the “full poly-isomorphism” of [i.e., the collection of all isomorphisms between] F ⊩ -prime-strips † F ⊩ tht ∼ → ‡ F ⊩ mod as the Θ-link from [the underlying Θ-Hodge theater of] † HT Θ±ell NF to [the underlying Θ-Hodge theater of] ‡ HT Θ±ell NF .TheΘ-link induces the full poly-isomorphism between the F ⊢× -prime-strips † F ⊢× mod ∼ → ‡ F ⊢× mod
associated to † F ⊩ mod and ‡ F ⊩ mod . INTER-UNIVERSAL TEICHMÜLLER THEORY I 15 (iii) (Frobenius-/Étale-Pictures) Let { n HT Θ±ell NF } n∈Z be a collection of distinct Θ ±ell NF-Hodge theaters [relative to the given initial Θ-data] indexed by the integers. Then the infinite chain ... Θ −→ (n−1) HT Θ±ell NF Θ −→ n HT Θ±ell NF Θ −→ (n+1) HT Θ±ell NF Θ −→ ... of Θ-linked Θ ±ell NF-Hodge theaters will be referred to as the Frobeniuspicture [associated to the Θ-link] — cf. Fig. I1.5. The Frobenius-picture fails to admit permutation automorphisms that switch adjacent indices n, n+1,but leave the remaining indices ∈ Z fixed. The Frobenius-picture induces an infinite chain of full poly-isomorphisms ... ∼ → (n−1) D ⊢ > ∼ → n D ⊢ > ∼ → (n+1) D ⊢ > ∼ → ... between the various D ⊢ -prime-strips n D ⊢ >, i.e., in essence, the D ⊢ -prime-strips associated to the F ⊢× -prime-strips n F ⊢× mod . The relationships of the various D- Θ ±ell NF-Hodge theaters n HT D-Θ±ellNF to the “mono-analytic core” constituted by the D ⊢ -prime-strip “ (−) D ⊢ >” regarded up to isomorphism — relationships that are depicted by spokes in Fig. I1.6 — are compatible with arbitrary permutation symmetries among the spokes [i.e., among the labels n ∈ Z of the D-Θ ±ell NF- Hodge theaters]. The diagram depicted in Fig. I1.6 will be referred to as the étalepicture. In addition to the main result discussed above, we also prove a certain technical result concerning tempered fundamental groups —cf. TheoremBbelow— that will be of use in our development of the theory of Hodge-Arakelov-theoretic evaluation in [IUTchII]. This result is essentially a routine application of the theory of maximal compact subgroups of tempered fundamental groups developed in [SemiAnbd] [cf., especially, [SemiAnbd], Theorems 3.7, 5.4]. Here, we recall that this theory of [SemiAnbd] may be thought of as a sort of “Combinatorial Section Conjecture” [cf. Remark 2.5.1 of the present paper; [IUTchII], Remark 1.12.4] — a point of view that is of particular interest in light of the historical remarks made in §I5 below. Moreover, Theorem B is of interest independently of the theory of the present series of papers in that it yields, for instance, a new proof of the normal terminality of the tempered fundamental group in its profinite completion, a result originally obtained in [André], Lemma 3.2.1, by means of other techniques [cf. Remark 2.4.1]. This new proof is of interest in that, unlike the techniques of [André], which are only available in the profinite case, this new proof [cf. Proposition 2.4, (iii)] holds in the case of pro-̂Σ-completions, for more general ̂Σ [i.e., not just the case of ̂Σ =Primes]. Theorem B. (Profinite Conjugates of Tempered Decomposition and Inertia Groups) Let k be a mixed-characteristic [nonarchimedean] local field, X a hyperbolic curve over k. Write Π tp X
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INTER-UNIVERSAL TEICHMÜLLER THEORY
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completely determined by † D. INT
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conjugation by which maps φ NF IN
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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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