Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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74 SHINICHI MOCHIZUKI Remark 3.7.1. One verifies immediately that there exist many distinct isomorphisms † F ⊩ ∼ tht → ‡ F ⊩ mod as in Corollary 3.7, (i), none of which is conferred a “distinguished” status, i.e., in the fashion of the “natural isomorphism F ⊩ ∼ mod → F ⊩ tht ” discussed in Example 3.5, (ii). The following result follows formally from Corollary 3.7. Corollary 3.8. (Frobenius-pictures of Θ-Hodge Theaters) Fix a collection of initial Θ-data as in Corollary 3.7. Let { n HT Θ } n∈Z be a collection of distinct Θ-Hodge theaters indexed by the integers. Then by applying Corollary 3.7, (i), with † HT Θ def = n HT Θ , ‡ HT Θ def = (n+1) HT Θ , we obtain an infinite chain ... Θ −→ (n−1) HT Θ Θ −→ n HT Θ Θ −→ (n+1) HT Θ Θ −→ ... of Θ-linked Θ-Hodge theaters. This infinite chain may be represented symbolically as an oriented graph ⃗ Γ [cf. [AbsTopIII], §0] ... → • → • → • → ... — i.e., where the arrows correspond to the “ −→’s”, Θ and the “•’s” correspond to the “ n HT Θ ”. This oriented graph ⃗ Γ admits a natural action by Z — i.e., a translation symmetry — but it does not admit arbitrary permutation symmetries. For instance, ⃗ Γ does not admit an automorphism that switches two adjacent vertices, but leaves the remaining vertices fixed. Put another way, from the point of view of the discussion of [FrdI], Introduction, the mathematical structure constituted by this infinite chain is “Frobenius-like”, or “order-conscious”. It is for this reason that we shall refer to this infinite chain in the following discussion as the Frobenius-picture. Remark 3.8.1. (i) Perhaps the central defining aspect of the Frobenius-picture is the fact that the Θ-link maps n Θ v ↦→ (n+1) q v [i.e., where v ∈ V bad — cf. the discussion of Example 3.2, (v)]. ... ---- n q v n Θ v ---- (n+1) q v (n+1) Θ v ---- ... n Θ v ↦→ (n+1) q v Fig. 3.1: Frobenius-picture of Θ-Hodge theaters
INTER-UNIVERSAL TEICHMÜLLER THEORY I 75 From this point of view, the Frobenius-picture may be depicted as in Fig. 3.1 — i.e., each box is a Θ-Hodge theater; the “” may be thought of as denoting the scheme theory that lies between “q v ”and“Θ v ”; the “- - - -” denotes the Θ-link. (ii) It is perhaps not surprising [cf. the theory of [FrdI]] that the Frobeniuspicture involves, in an essential way, the divisor monoid portion [i.e., “q ”and v “Θ v ”] of the various Frobenioids that appear in a Θ-Hodge theater. Put another way, it is as if the “Frobenius-like nature” of the divisor monoid portion of the Frobenioids involved induces the “Frobenius-like nature” of the Frobeniuspicture. By contrast, observe that for v ∈ V, the isomorphisms ... ∼ → n D ⊢ v ∼ → (n+1) D ⊢ v ∼ → ... of Corollary 3.7, (ii), imply that if one thinks of the various (−) Dv ⊢ known up to isomorphism, then as being only one may regard (−) Dv ⊢ as a sort of constant invariant of the various Θ-Hodge theaters that constitute the Frobenius-picture — cf. Remark 3.9.1 below. This observation is the starting point of the theory of the étale-picture [cf. Corollary 3.9, (i), below]. Note that by Corollary 3.7, (iii), we also obtain isomorphisms ... ∼ → O ×n C ⊢ v ∼ → O × (n+1) C ⊢ v ∼ → ... lying over the isomorphisms involving the “ (−) D ⊢ v ” discussed above. (iii) In the situation of (ii), suppose that v ∈ V non . Then (−) Dv ⊢ is simply the category of connected objects of the Galois category associated to the profinite group G v . That is to say, one may think of (−) Dv ⊢ as representing “G v up to isomorphism”. Then each n D v represents an “isomorph of the topological group Π v , labeled by n, which is regarded as an extension of some isomorph of G v that is independent of n”. In particular, the quotients corresponding to G v of the copies of Π v that arise from n HT Θ for different n are only related to one another via some indeterminate isomorphism. Thus, from the point of view of the theory of [AbsTopIII] [cf. [AbsTopIII], §I3; [AbsTopIII], Remark 5.10.2, (ii)], each Π v gives rise to a well-defined ring structure — i.e., a “holomorphic structure” —whichis obliterated by the indeterminate isomorphism between the quotient isomorphs of G v arising from n HT Θ for distinct n. (iv) In the situation of (ii), suppose that v ∈ V arc . Then (−) Dv ⊢ is an object of TM ⊢ ; each n D v represents an “isomorph of the Aut-holomorphic orbispace −→v X , labeled by n, whose associated [complex archimedean] topological field A X−→v gives
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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