Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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102 SHINICHI MOCHIZUKI (ii) Since the F l -symmetry that appears in Proposition 4.9, (i), is transitive, it follows that one may use this action to perform comparisons as discussed in (i). This prompts the question: What is the difference between this F l -symmetry and the S l -symmetry of the output data of the functors of Proposition 4.9, (ii), (iii)? Inaword,restrictingtotheF l -symmetry of Proposition 4.9, (i), amounts to the imposition of a “cyclic structure” on the index set J [i.e., a structure of F l -torsor on J]. Thus, relative to the issue of comparability raised in (i), this F l -symmetry allows comparison between — i.e., involves isomorphisms between the non-labeled D-prime-strips corresponding to — distinct members of this index set J, without disturbing the cyclic structure on J. This cyclic structure may be thought of as a sort of combinatorial manifestation of the link to the global object † D ⊚ that appears in a D-NF-bridge. On the other hand, in order to compare these D-prime-strips indexed by J “in the absolute” to D-prime-strips that have nothing to do with J, it is necessary the “forget the cyclic structure on J”. This is precisely what is achieved by considering the functors of Proposition 4.9, (ii), (iii), i.e., by working with the “full S l -symmetry”. Remark 4.9.2. (i) The various elements of the index set of the capsule of D-prime-strips of a D-NF-bridge are synchronized in their correspondence with the labels “1, 2,... ,l ”, in the sense that this correspondence is completely determined up to composition withtheactionofanelementofF l . In particular, this correspondence is always bijective. One may regard this phenomenon of synchronization, orcohesion, as an important consequence of the fact that the number field in question is represented in the D-NF-bridge via a single copy [i.e., as opposed to a capsule whose index set is of cardinality ≥ 2] of D ⊚ . Indeed, consider a situation in which each D-prime-strip in the capsule † D J is equipped with its own “independent globalization”, i.e., copy of D ⊚ ,towhichit is related by a copy of “φ NF j ”, which [in order not to invalidate the comparability of distinct labels — cf. Remark 4.9.1, (i)] is regarded as being known only up to composition with the action of an element of F l . Then if one thinks of the [manifestly mutually disjoint — cf. Definition 3.1, (f); Example 4.3, (i)] F l -translates of V ±un ⋂ V(K) bad [whose union is equal to V Bor ⋂ V(K) bad ]asbeinglabeled by the elements of F l ,theneach D-prime-strip in the capsule † D J — i.e., each “•” in Fig. 4.5 below — is subject, as depicted in Fig. 4.5, to an independent indeterminacy concerning the label ∈ F l to which it is associated. In particular, the set of all possibilities for each association includes correspondences between the index set J of the capsule † D J and the set of labels F l which fail to be bijective. Moreover,
INTER-UNIVERSAL TEICHMÜLLER THEORY I 103 although F l arises essentially as a subquotient of a Galois group of extensions of number fields [cf. the faithful poly-action of F l on primes of V(K)], the fact that it also acts faithfully on conjugates of the cusp ɛ [cf. Example 4.3, (i)] implies that “working with elements of V(K) up to F l -indeterminacy” may only be done at the expense of “working with conjugates of the cusp ɛ up to F l -indeterminacy”. Thatis to say, “working with nonsynchronized labels” is inconsistent with the construction of the crucial bijection † ζ in Proposition 4.7, (iii). • ↦→ 1? 2? 3? ··· l ? • ↦→ 1? 2? 3? ··· l ? . . • ↦→ 1? 2? 3? ··· l ? Fig. 4.5: Nonsynchronized labels (ii) In the context of the discussion of (i), we observe that the “single copy” of D ⊚ may also be thought of as a “single connected component”, hence — from the point of view of Galois categories —asa“single basepoint”. (iii) In the context of the discussion of (i), it is interesting to note that since the natural action of F l on F l is transitive, one obtains the same “set of all possibilities for each association”, regardless of whether one considers independent F l -indeterminacies at each index of J or independent S l-indeterminacies at each index of J [cf. the discussion of Remark 4.9.1, (ii)]. (iv) The synchronized indeterminacy [cf. (i)] exhibited by a D-NF-bridge — i.e., at a more concrete level, the crucial bijection † ζ of Proposition 4.7, (iii) — may be thought of as a sort of combinatorial model of the notion of a “holomorphic structure”. By contrast, the nonsynchronized indeterminacies discussed in (i) may be thought of as a sort of combinatorial model of the notion of a “real analytic structure”. Moreover, we observe that the theme of the above discussion — in which one considers “how a given combinatorial holomorphic structure is ‘embedded’ within its underlying combinatorial real analytic structure” — is very much in line with the spirit of classical complex Teichmüller theory. (v) From the point of view discussed in (iv), the main results of the “multiplicative combinatorial Teichmüller theory” developed in the present §4 may be summarized as follows: (a) globalizability of labels, in a fashion that is independent of local structures [cf. Remark 4.3.2, (b); Proposition 4.7, (iii)]; (b) comparability of distinct labels [cf. Proposition 4.9; Remark 4.9.1, (i)]; (c) absolute comparability [cf. Proposition 4.9, (ii), (iii); Remark 4.9.1, (ii)]; (d) minimization of label indeterminacy — without sacrificing the symmetry necessary to perform comparisons! — via processions [cf. Proposition 4.11, (i), (ii), below].
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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