Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

More documents

Recommendations

Info

76 SHINICHI MOCHIZUKI rise to an isomorph of Dv ⊢ that is independent of n”. In particular, the various isomorphs of Dv ⊢ associated to the copies of −→v X 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 −→v X gives rise to a well-defined ring structure — i.e., a “holomorphic structure” —whichisobliterated by the indeterminate isomorphism between the isomorphs of Dv ⊢ arising from n HT Θ for distinct n. The discussion of Remark 3.8.1, (iii), (iv), may be summarized as follows. Corollary 3.9. (Étale-pictures of Θ-Hodge Theaters) In the situation of Corollary 3.8, let v ∈ V. Then: n D v ... | ... | (n−1) D v — — D ⊢ v — — (n+1) D v ... | | ... (n+2) D v Fig. 3.2: Étale-picture of Θ-Hodge theaters (i) We have a diagram as in Fig. 3.2, which we refer to as the étale-picture. Here, each horizontal and vertical “— —” denotes the relationship between (−) D v and Dv ⊢ — i.e., an extension of topological groups when v ∈ V non , or the underlying object of TM ⊢ arising from the associated topological field when v ∈ V arc —discussed in Remark 3.8.1, (iii), (iv). [Unlike the Frobenius-picture!] the étale-picture admits arbitrary permutation symmetries among the labels n ∈ Z corresponding to the various Θ-Hodge theaters. Put another way, the étale-picture may be thought of as a sort of canonical splitting of the Frobenius-picture. (ii) In a similar vein, we have a diagram as in Fig. 3.3 below, obtained by replacing the “Dv ⊢ ” in the middle of Fig. 3.2 by “Dv ⊢ O × ”. Here, each Cv ⊢ horizontal and vertical “— —” denotes the relationship between (−) D v and Dv ⊢ discussed in (i); when v ∈ V non , the notation “Dv ⊢ O × ” denotes an isomorph Cv ⊢ of the pair consisting of the category D ⊢ v together with the group-like monoid O × C ⊢ v
INTER-UNIVERSAL TEICHMÜLLER THEORY I 77 on D ⊢ v ;whenv ∈ V arc , the notation “D ⊢ v O × C ⊢ v ” denotes an isomorph of the pair consisting of the object Dv ⊢ ∈ Ob(TM ⊢ ) and the topological group O × [which Cv ⊢ is isomorphic — but not canonically! — to the compact factor of Dv ⊢ ]. Just as in the case of (i), this diagram admits arbitrary permutation symmetries among the labels n ∈ Z corresponding to the various Θ-Hodge theaters. n D v ... | ... | (n−1) D v — — D ⊢ v O × C ⊢ v — — (n+1) D v ... | | ... (n+2) D v Fig. 3.3: Étale-picture plus units Remark 3.9.1. If one formulates things relative to the language of [AbsTopIII], Definition 3.5, then (−) Dv ⊢ constitutes a core. Relative to the theory of [AbsTopIII], §5, this core is essentially the mono-analytic core discussed in [AbsTopIII], §I3; [AbsTopIII], Remark 5.10.2, (ii). Indeed, the symbol “⊢” is intended — both in [AbsTopIII] and in the present series of papers! — as an abbreviation for the term “mono-analytic”. Remark 3.9.2. Whereas the étale-picture of Corollary 3.9, (i), will remain valid throughout the development of the remainder of the theory of the present series of papers, the local units “O × ” that appear in Corollary 3.9, (ii), will ultimately cease Cv ⊢ to be a constant invariant of various enhanced versions of the Frobenius-picture that will arise in the theory of [IUTchIII]. In a word, these enhancements revolve around the incorporation into each Hodge theater of the “rotation of addition (i.e., ‘⊞’) and multiplication (i.e., ‘⊠’)” in the style of the theory of [AbsTopIII]. Remark 3.9.3. (i) As discussed in [AbsTopIII], §I3; [AbsTopIII], Remark 5.10.2, (ii), the “mono-analytic core” {D ⊢ v } v∈V may be thought of as a sort of fixed underlying real-analytic surface associated to a number field on which various holomorphic structures are imposed. Then the Frobenius-picture in its entirety may
Page 1 and 2:
INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 3 and 4:
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
associated to † F ⊩ mod and ‡
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 75: INTER-UNIVERSAL TEICHMÜLLER THEORY
Page 81 and 82: completely determined by † D. INT
Page 97 and 98: conjugation by which maps φ NF IN
Page 127 and 128:
Page 129 and 130:
(v) Write INTER-UNIVERSAL TEICHMÜL
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157:
show all

Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?