Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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24 SHINICHI MOCHIZUKI combinatorial dimensions” of a ring that leave the arithmetic holomorphic structure fixed — cf. the discussion of the “juggling of ⊞, ⊠ induced by log” in [AbsTopIII], §I3. The ultimate conclusion of the theory of [IUTchIII] is that the “a priori unbounded deformations” of the arithmetic holomorphic structure given by the Θ-link in fact admit canonical bounds, which may be thought of as a sort of reflection of the “hyperbolicity” of the given number field equipped with an elliptic curve — cf. [IUTchIII], Corollary 3.12. Such canonical bounds may be thought of as analogues for a number field of canonical bounds that arise from differentiating Frobenius liftings in the context of p-adic hyperbolic curves — cf. the discussion in the final portion of [AbsTopIII], §I5. Moreover, such canonical bounds are obtained in [IUTchIII] as a consequence of the explicit description of a varying arithmetic holomorphic structure within a fixed mono-analytic “container” — cf. the discussion of §I2! — furnished by [IUTchIII], Corollary 3.11 [cf. also the discussion of [IUTchIII], Remarks 3.12.2, 3.12.3, 3.12.4], i.e., a situation that is entirely formally analogous to the summary of complex Teichmüller theory given above. The significance of the log-theta-lattice is best understood in the context of the analogy between the inter-universal Teichmüller theory developedinthe present series of papers and the p-adic Teichmüller theory of [pOrd], [pTeich]. Here, we recall for the convenience of the reader that the p-adic Teichmüller theory of [pOrd], [pTeich] may be summarized, [very!] roughly speaking, as a sort of generalization, to the case of “quite general” p-adic hyperbolic curves, of the classical p-adic theory surrounding the canonical representation π 1 ((P 1 \{0, 1, ∞}) Qp ) → π 1 ((M ell ) Qp ) → PGL 2 (Z p ) —wherethe“π 1 (−)’s” denote the étale fundamental group, relative to a suitable basepoint; (M ell ) Qp denotes the moduli stack of elliptic curves over Q p ; the first horizontal arrow denotes the morphism induced by the elliptic curve over the projective line minus three points determined by the classical Legendre form of the Weierstrass equation; the second horizontal arrow is the representation determined by the p-power torsion points of the tautological elliptic curve over (M ell ) Qp . In particular, the reader who is familiar with the theory of the classical representation of the above display, but not with the theory of [pOrd], [pTeich], may nevertheless appreciate, to a substantial degree, the analogy between the inter-universal Teichmüller theory developed in the present series of papers and the p-adic Teichmüller theory of [pOrd], [pTeich] by thinking in terms of the well-known classical properties of this classical representation. In some sense, the gap between the “quite general” p-adic hyperbolic curves that appear in p-adic Teichmüller theory and the classical case of (P 1 \{0, 1, ∞}) Qp may
INTER-UNIVERSAL TEICHMÜLLER THEORY I 25 be thought of, roughly speaking, as corresponding, relative to the analogy with the theory of the present series of papers, to the gap between arbitrary number fields and the rational number field Q. This point of view is especially interesting in the context of the discussion of §I5 below. Inter-universal Teichmüller theory p-adic Teichmüller theory number field F hyperbolic curve C over a positive characteristic perfect field [once-punctured] elliptic curve X over F nilpotent ordinary indigenous bundle P over C Θ-link arrows of the log-theta-lattice mixed characteristic extension structure of a ring of Witt vectors log-link arrows of the log-theta-lattice the Frobenius morphism in positive characteristic the entire log-theta-lattice the resulting canonical lifting + canonical Frobenius action; canonical Frobenius lifting over the ordinary locus relatively straightforward original construction of log-theta-lattice relatively straightforward original construction of canonical liftings highly nontrivial description of alien arithmetic holomorphic structure via absolute anabelian geometry highly nontrivial absolute anabelian reconstruction of canonical liftings Fig. I4.1: Correspondence between inter-universal Teichmüller theory and p-adic Teichmüller theory The analogy between the inter-universal Teichmüller theory developed in the present series of papers and the p-adic Teichmüller theory of [pOrd], [pTeich]
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conjugation by which maps φ NF IN
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(v) Write INTER-UNIVERSAL TEICHMÜL
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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