Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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98 SHINICHI MOCHIZUKI Proposition 4.2. Assertions (ii) and (iii) follow immediately from the intrinsic nature of the constructions of Example 4.5. ○ Remark 4.7.1. The significance of the natural bijection † ζ of Proposition 4.7, (iii), lies in the following observation: Suppose that one wishes to work with the global data † D ⊚ in a fashion that is independent of the local data [i.e., “prime-strip data”] † D > , † D J [cf. Remark 4.3.2, (b)]. Then by replacing the capsule index set J by the set of global label classes of cusps LabCusp( † D ⊚ ) via † ζ , one obtains an object — i.e., LabCusp( † D ⊚ ) — constructed via [i.e., “native to”] the global data that is immune to the “collapsing” of J → ∼ F l — i.e., of F l -orbits of V±un — even at primes v ∈ V of the sort discussed in Remark 4.2.1! That is to say, this “collapsing” of [i.e., failure of F l to act freely on] F l -orbits of V ±un is a characteristically global consequence of the global prime decomposition trees discussed in Remark 4.3.1, (ii) [cf. the example discussed in Remark 4.2.1]. We refer to Remark 4.9.3, (ii), below for a discussion of a closely related phenomenon. Remark 4.7.2. (i)Attheleveloflabels [cf. the content of Proposition 4.7], the structure of a D-ΘNF-Hodge theater may be summarized via the diagram of Fig. 4.4 below — i.e., where the expression “[ 1 < 2 < ... < (l − 1) < l ]” corresponds to † D > ;the expression “( 1 2 ... l − 1 l )” corresponds to † D J ; the lower righthand “F l -cycle of ’s” corresponds to † D ⊚ ;the“⇑” corresponds to the associated D-Θ-bridge; the“⇒” corresponds to the associated D-NF-bridge; the“/ ’s” denote D-prime-strips. (ii) Note that the labels arising from † D > correspond, ultimately, to various irreducible components in the special fiber of a certain tempered covering of a [“geometric”!] Tate curve [a special fiber which consists of a chain of copies of the projective line — cf. [EtTh], Corollary 2.9]. In particular, these labels are obtained by counting —inanintuitive,archimedean, additive fashion — the number of irreducible components between a given irreducible component and the “origin”. In particular, the portion of the diagram of Fig. 4.4 corresponding to † D > may be described by the following terms: geometric, additive, archimedean, hence Frobenius-like [cf. Corollary 3.8]. By contrast, the various “’s” in the portion of the diagram of Fig. 4.4 corresponding to † D ⊚ arise, ultimately, from various primes of an [“arithmetic”!] number field. These primes are permuted by the multiplicative group F l = F × l /{±1}, ina cyclic — i.e., nonarchimedean — fashion. Thus, the portion of the diagram of Fig. 4.4 corresponding to † D ⊚ may be described by the following terms: arithmetic, multiplicative, nonarchimedean, hence étale-like [cf. the discussion of Remark 4.3.2].
INTER-UNIVERSAL TEICHMÜLLER THEORY I 99 That is to say, the portions of the diagram of Fig. 4.4 corresponding to † D > , † D ⊚ differ quite fundamentally in structure. In particular, it is not surprising that the only “common ground” of these two fundamentally structurally different portions consists of an underlying set of cardinality l [i.e., the portion of the diagram of Fig. 4.4 corresponding to † D J ]. (iii) The bijection † ζ — or, perhaps more appropriately, its inverse ( † ζ ) −1 : J ∼ → LabCusp( † D ⊚ ) — may be thought of as relating arithmetic [i.e., if one thinks of the elements of the capsule index set J as collections of primes of a number field] togeometry [i.e., if one thinks of the elements of LabCusp( † D ⊚ ) as corresponding to the [geometric!] cusps of the hyperbolic orbicurve]. From this point of view, ( † ζ ) −1 may be thought of as a sort of “combinatorial Kodaira-Spencer morphism” [cf. the point of view of [HASurI], §1.4]. We refer to Remark 4.9.2, (iv), below, for another way to think about † ζ . [1< 2
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associated to † F ⊩ mod and ‡
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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