Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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108 SHINICHI MOCHIZUKI Section 5: ΘNF-Hodge Theaters In the present §5, we continue our discussion of various “enhancements” to the Θ-Hodge theaters of §3. Namely, we define the notion of a ΘNF-Hodge theater [cf. Definition 5.5, (iii)] and observe that these ΘNF-Hodge theaters satisfy the same “functorial dynamics” [cf. Corollary 5.6; Remark 5.6.1] as the base-ΘNF- Hodge theaters discussed in §4. Let † HT D-ΘNF =( † D ⊚ † φ NF ←− † † φ Θ D J −→ † D > ) be a D-ΘNF-Hodge theater [cf. Definition 4.6], relative to a fixed collection of initial Θ-data (F/F, X F , l, C K , V, V bad mod , ɛ) as in Definition 3.1. Example 5.1. Global Frobenioids. (i) By applying the anabelian result of [AbsTopIII], Theorem 1.9, via the “Θ-approach” discussed in Remark 3.1.2, to π 1 ( † D ⊚ ), we may construct grouptheoretically from π 1 ( † D ⊚ )anisomorphof“F × ” — which we shall denote M ⊚ ( † D ⊚ ) — equipped with its natural π 1 ( † D ⊚ )-action. Here, we recall that this construction includes a reconstruction of the field structure on M ⊚ ( † D ⊚ ) def = M ⊚ ( † D ⊚ ) ⋃ {0}. Next, let us observe that the F -core C F [cf. [CanLift], Remark 2.1.1; [EtTh], the discussion at the beginning of §2] admits a unique model C Fmod over F mod . In particular, it follows that one may construct group-theoretically from π 1 ( † D ⊚ ), in a functorial fashion, a profinite group corresponding to “C Fmod ” [cf. the algorithms of [AbsTopII], Corollary 3.3, (i), which are applicable in light of [AbsTopI], Example 4.8; the definition of “F mod ” in Definition 3.1, (b)], which contains π 1 ( † D ⊚ )asan open subgroup; write † D ⊛ for B(−) 0 of this profinite group, so we obtain a natural morphism † D ⊚ → † D ⊛ — i.e., a “category-theoretic version” of the natural morphism of hyperbolic orbicurves C K → C Fmod — together with a natural extension of the action of π 1 ( † D ⊚ ) on M ⊚ ( † D ⊚ )toπ 1 ( † D ⊛ ). In particular, by taking π 1 ( † D ⊛ )-invariants, we obtain a submonoid/subfield M ⊚ mod († D ⊚ ) ⊆ M ⊚ ( † D ⊚ ), M ⊚ mod( † D ⊚ ) ⊆ M ⊚ ( † D ⊚ ) corresponding to F × mod ⊆ F × , F mod ⊆ F . (ii) Next, let us recall [cf. Definition 4.1, (v)] that the field structure on M ⊚ ( † D ⊚ ) [i.e., “F ”] allows one to reconstruct group-theoretically from π 1 ( † D ⊚ ) the set of valuations V( † D ⊚ ) [i.e., “V(F )”] on M ⊚ ( † D ⊚ ) equipped with its natural
INTER-UNIVERSAL TEICHMÜLLER THEORY I 109 π 1 ( † D ⊛ )-action, hence also the monoid on † D ⊛ [i.e., in the sense of [FrdI], Definition 1.1, (ii)] Φ ⊚ ( † D ⊚ )(−) that associates to an object A ∈ Ob( † D ⊛ )themonoid Φ ⊚ ( † D ⊚ )(A) of “stacktheoretic” [cf. Remark 3.1.5] arithmetic divisors on the corresponding subfield M ⊚ ( † D ⊚ ) A ⊆ M ⊚ ( † D ⊚ ) [i.e., the monoid denoted “Φ(−)” in [FrdI], Example 6.3; cf. also Remark 3.1.5 of the present paper], together with the natural morphism of monoids M ⊚ ( † D ⊚ ) A → Φ ⊚ ( † D ⊚ )(A) gp [cf. the discussion of [FrdI], Example 6.3; Remark 3.1.5 of the present paper]. As discussed in [FrdI], Example 6.3 [cf. also Remark 3.1.5 of the present paper], this data determines, by applying [FrdI], Theorem 5.2, (ii), a model Frobenioid over the base category † D ⊛ . F ⊛ ( † D ⊚ ) (iii) Let † F ⊛ be any category equivalent to F ⊛ ( † D ⊚ ). Thus, † F ⊛ is equipped with a natural Frobenioid structure [cf. [FrdI], Corollary 4.11; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper]; write Base( † F ⊛ ) for the base category of this Frobenioid. Suppose further that we have been given a morphism † D ⊚ → Base( † F ⊛ ) which is abstractly equivalent [cf. §0] to the natural morphism † D ⊚ → † D ⊛ [cf. (i)]. In the following discussion, we shall use the resulting [uniquely determined, in light of the F -coricity of C F , together with [AbsTopIII], Theorem 1.9!] isomorphism Base( † F ⊛ ) → ∼ † D ⊛ to identify Base( † F ⊛ )with † D ⊛ . Let us denote by † F ⊚ def = † F ⊛ |† D ⊚ (⊆ † F ⊛ ) the restriction of † F ⊛ to † D ⊚ via the natural morphism † D ⊚ → † D ⊛ and by † F ⊚ mod def = † F ⊛ | terminal objects (⊆ † F ⊛ ) the restriction of † F ⊛ to the full subcategory of † D ⊛ determined by the terminal objects [i.e., “C Fmod ”] of † D ⊛ . Thus, when the data denoted here by the label “†” arises [in the evident sense] from data as discussed in Definition 3.1, the Frobenioid † F ⊚ mod may be thought of as the Frobenioid of arithmetic line bundles on the stack “S mod ” of Remark 3.1.5. (iv) We continue to use the notation of (iii). We shall denote by a superscript “birat” the birationalizations [which are category-theoretic — cf. [FrdI], Corollary 4.10; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper] of the Frobenioids † F ⊚ , † F ⊛ , † F ⊚ mod ; we shall also use this superscript to denote the images of objects and morphisms of these Frobenioids in their birationalizations. Thus, if A ∈ Ob( † F ⊛ ), then O × (A birat ) may be naturally identified with the multiplicative group of nonzero elements of the number field [i.e., finite extension of F mod ] corresponding to A. In particular, by allowing A to vary among the Frobenius-trivial
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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