Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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16 SHINICHI MOCHIZUKI for the tempered fundamental group π tp 1 (X) [relative to a suitable basepoint] of X [cf. [André], §4; [SemiAnbd], Example 3.10]; ̂Π X for the étale fundamental group [relative to a suitable basepoint] of X. Thus, we have a natural inclusion Π tp X ↩→ ̂Π X which allows one to identify ̂Π X with the profinite completion of Π tp X . Then every decomposition group in ̂Π X (respectively, inertia group in ̂Π X ) associated to a closed point or cusp of X (respectively, to a cusp of X) iscontainedinΠ tp X if and only if it is a decomposition group in Π tp X (respectively, inertia group in Πtp X ) associated to a closed point or cusp of X (respectively, to a cusp of X). Moreover, a ̂Π X -conjugate of Π tp X contains a decomposition group in Πtp X (respectively, inertia group in Π tp X ) associated to a closed point or cusp of X (respectively, to a cusp of X) if and only if it is equal to Π tp X . Theorem B is [essentially] given as Corollary 2.5 in §2. Here, we note that although, in the statement of Corollary 2.5, the hyperbolic curve X is assumed to admit stable reduction over the ring of integers O k of k, one verifies immediately that this assumption is, in fact, unnecessary. Finally, we remark that one important reason for the need to apply Theorem B in the context of the theory of Θ ±ell NF-Hodge theaters summarized in Theorem A is the following. The F ⋊± l -symmetry, which will play a crucial role in the theory of the present series of papers [cf., especially, [IUTchII], [IUTchIII]], depends, in an essential way, on the synchronization of the ±-indeterminacies that occur locally at each v ∈ V [cf. Fig. I1.1]. Such a synchronization may only be obtained by making use of the global portion of the Θ ±ell -Hodge theater under consideration. On the other hand, in order to avail oneself of such global ±-synchronizations [cf. Remark 6.12.4, (iii)], it is necessary to regard the various labels of the F ⋊± l - symmetry ( −l < ... < −1 < 0 < 1 < ... < l ) as conjugacy classes of inertia groups of the [necessarily] profinite geometric étale fundamental group of X K . That is to say, in order to relate such global profinite conjugacy classes to the corresponding tempered conjugacy classes [i.e., conjugacy classes with respect to the geometric tempered fundamental group] of inertia groups at v ∈ V bad [i.e., where the crucial Hodge-Arakelov-theoretic evaluation is to be performed!], it is necessary to apply Theorem B — cf. the discussion of Remark 4.5.1; [IUTchII], Remark 2.5.2, for more details. §I2. Gluing Together Models of Conventional Scheme Theory As discussed in §I1, the system of Frobenioids constituted by a Θ ±ell NF-Hodge theater is intended to be a sort of miniature model of conventional scheme theory. Onethenglues multiple Θ ±ell NF-Hodge theaters { n HT Θ±ell NF } n∈Z together
INTER-UNIVERSAL TEICHMÜLLER THEORY I 17 by means of the full poly-isomorphisms between the “subsystems of Frobenioids” constituted by certain F ⊩ -prime-strips † F ⊩ tht ∼ → ‡ F ⊩ mod to form the Frobenius-picture. One fundamental observation in this context is the following: these gluing isomorphisms — i.e., in essence, the correspondences n Θ v ↦→ (n+1) q v — and hence the geometry of the resulting Frobenius-picture lie outside the framework of conventional scheme theory in the sense that they do not arise from ring homomorphisms! In particular, although each particular model n HT Θ±ell NF of conventional scheme theory is constructed within the framework of conventional scheme theory, the relationship between the distinct [albeit abstractly isomorphic, as Θ ±ell NF-Hodge theaters!] conventional scheme theories represented by, for instance, neighboring Θ ±ell NF-Hodge theaters n HT Θ±ell NF , n+1 HT Θ±ell NF cannot be expressed schemetheoretically. In this context, it is also important to note that such gluing operations are possible precisely because of the relatively simple structure — for instance, by comparison to the structure of a ring! — of the Frobenius-like structures constituted by the Frobenioids that appear in the various F ⊩ -prime-strips involved, i.e., in essence, collections of monoids isomorphic to N or R ≥0 [cf. Fig. I1.2]. Fig. I2.1: Depiction of Frobenius- and étale-pictures of Θ ±ell NF-Hodge theaters via glued topological surfaces
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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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