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Inter-Universal Teichmüller Theory III: Canonical Splittings of the Log

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Inter-Universal Teichmüller Theory III: Canonical Splittings of the Log INTER-UNIVERSAL TEICHMULLER¨ THEORY III: CANONICAL SPLITTINGS OF THE LOG-THETA-LATTICE Shinichi Mochizuki May 2020 Abstract. The present paper constitutes the third paper in a series of four papers and may be regarded as the culmination of the abstract conceptual por- tion of the theory developed in the series. In the present paper, we study the theory surrounding the log-theta-lattice,ahighly non-commutative two-dimensional dia- gram of “miniature models of conventional scheme theory”, called Θ±ellNF-Hodge theaters. Here, we recall that Θ±ellNF-Hodge theaters were associated, in the first paper of the series, to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Each ar- row of the log-theta-lattice corresponds to a certain gluing operation between the Θ±ellNF-Hodge theaters in the domain and codomain of the arrow. The horizontal arrows of the log-theta-lattice are defined as certain versions of the “Θ-link” that was constructed, in the second paper of the series, by applying the theory of Hodge- Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the l-th root of the] theta function at l-torsion points.Inthe present paper, we focus on the theory surrounding the log-link between Θ±ellNF- Hodge theaters. The log-link is obtained, roughly speaking, by applying, at each [say, for simplicity, nonarchimedean] valuation of the number field under consider- ation, the local p-adic logarithm. The significance of the log-link lies in the fact that it allows one to construct log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local units at the valuation under consideration via the local p-adic logarithm. The theory of log-shells was studied extensively in a previ- ous paper by the author. The vertical arrows of the log-theta-lattice are given by the log-link. Consideration of various properties of the log-theta-lattice leads natu- rally to the establishment of multiradial algorithms for constructing “splitting monoids of logarithmic Gaussian procession monoids”. Here, we recall that “multiradial algorithms” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. These loga- rithmic Gaussian procession monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of the Gaussian monoids that were studied in the second paper of the series. Finally, by applying these multiradial algorithms for splitting monoids of LGP-monoids, we obtain estimates for the log-volume of these LGP-monoids. Explicit computations of these estimates will be applied, in the fourth paper of the series, to derive various diophantine results. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI Contents: Introduction §0. Notations and Conventions §1. The Log-theta-lattice §2. Multiradial Theta Monoids §3. Multiradial Logarithmic Gaussian Procession Monoids Introduction In the following discussion, we shall continue to use the notation of the In- troduction to the first paper of the present series of papers [cf. [IUTchI], §I1]. In particular, we assume that are given an elliptic curve EF over a number field F ,to- gether with a prime number l ≥ 5. In the first paper of the series, we introduced and studied the basic properties of Θ±ellNF-Hodge theaters, which may be thought of as miniature models of the conventional scheme theory surrounding the given elliptic curve EF over the number field F . In the present paper, which forms the third paper of the series, we study the theory surrounding the log-link between Θ±ellNF-Hodge theaters. The log-link induces an isomorphism between the underlying D-Θ±ellNF- Hodge theaters and, roughly speaking, is obtained by applying, at each [say, for simplicity, nonarchimedean] valuation v ∈ V,thelocal pv-adic logarithm to the lo- cal units [cf. Proposition 1.3, (i)]. The significance of the log-link lies in the fact that it allows one to construct log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local units at v ∈ V via the local pv-adic logarithm. The theory of log-shells was studied extensively in [AbsTopIII]. The introduction of log-shells leads naturally to the construction of new versions — namely, the ×μ ×μ ×μ ×μ ΘLGP-/Θlgp -links [cf. Definition 3.8, (ii)] — of the Θ-/Θ -/Θgau-links studied in [IUTchI], [IUTchII]. The resulting [highly non-commutative!] diagram of iterates ×μ ×μ ×μ ×μ of the log- [i.e., the vertical arrows]andΘ -/Θgau-/ΘLGP-/Θlgp -links [i.e., the horizontal arrows] — which we refer to as the log-theta-lattice [cf. Definitions ×μ 1.4; 3.8, (iii), as well as Fig. I.1 below, in the case of the ΘLGP-link] — plays a central role in the theory of the present series of papers. ⏐ ⏐ ⏐log ⏐log ×μ ×μ ×μ Θ ±ell Θ ±ell Θ ... −→LGP n,m+1HT Θ NF −→LGP n+1,m+1HT Θ NF −→LGP ... ⏐ ⏐ ⏐log ⏐log ×μ ×μ ×μ Θ ±ell Θ ±ell Θ ... −→LGP n,mHT Θ NF −→LGP n+1,mHT Θ NF −→LGP ... ⏐ ⏐ ⏐log ⏐log . Fig. I.1: The [LGP-Gaussian] log-theta-lattice INTER-UNIVERSAL TEICHMULLER¨ THEORY III 3 Consideration of various properties of the log-theta-lattice leads naturally to the establishment of multiradial algorithms for constructing “splitting monoids of logarithmic Gaussian procession monoids” [cf. Theorem A below]. Here, we recall that “multiradial algorithms” [cf. the discussion of [IUTchII], Introduc- tion] are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ±ellNF-Hodge theater related to a given Θ±ellNF-Hodge theater by means of a non-ring/scheme- ×μ ×μ ×μ ×μ theoretic Θ-/Θ -/Θgau-/ΘLGP-/Θlgp -link. These logarithmic Gaussian procession monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of the Gaussian monoids that were studied in [IUTchII]. Finally, by apply- ing these multiradial algorithms for splitting monoids of LGP-monoids, we obtain estimates for the log-volume of these LGP-monoids [cf. Theorem B below]. These estimates will be applied to verify various diophantine results in [IUTchIV]. Recall [cf. [IUTchI], §I1] the notion of an F-prime-strip.AnF-prime-strip consists of data indexed by the valuations v ∈ V; roughly speaking, the data at each v consists of a Frobenioid, i.e., in essence, a system of monoids over a base category. For instance, at v ∈ Vbad, this data may be thought of as an isomorphic copy of the monoid with Galois action Πv O F v — where we recall that O denotes the multiplicative monoid of nonzero integral F v elements of the completion of an algebraic closure F of F at a valuation lying over × v [cf. [IUTchI], §I1, for more details]. The pv-adic logarithm logv : O → F v at v F v then defines a natural Πv-equivariant isomorphism of ind-topological modules ×μ ∼ × ∼ (O ⊗ Q → ) O ⊗ Q → F v F v F v ×μ μ — where we recall the notation “O = O× /O ” from the discussion of [IUTchI], F v F v F v §1 — which allows one to equip O× ⊗ Q with the field structure arising from the F v field structure of F v.Theportionatv of the log-link associated to an F-prime-strip [cf. Definition 1.1, (iii); Proposition 1.2] may be thought of as the correspondence log Πv O −→ Πv O F v F v in which one thinks of the copy of “O ”ontheright as obtained from the field F v structure induced by the pv-adic logarithm on the tensor product with Q of the copy of the units “O× ⊆O ”ontheleft. Since this correspondence induces an F v F v isomorphism of topological groups between the copies of Πv on either side, one may think of Πv as “immune to”/“neutral with respect to” — or, in the terminology of the present series of papers, “coric” with respect to — the transformation constituted by the log-link. This situation is studied in detail in [AbsTopIII], §3, and reviewed in Proposition 1.2 of the present paper. By applying various results from absolute anabelian geometry,onemay algorithmically reconstruct a copy of the data “Πv O ”fromΠv.Moreover, F v 4 SHINICHI MOCHIZUKI by applying Kummer theory, one obtains natural isomorphisms between this “coric version” of the data “Πv O ” and the copies of this data that appear on F v either side of the log-link. On the other hand, one verifies immediately that these Kummer isomorphisms are not compatible with the coricity of the copy of the data “Πv O ” algorithmically constructed from Πv. This phenomenon is, in F v some sense, the central theme of the theory of [AbsTopIII], §3, and is reviewed in Proposition 1.2, (iv), of the present paper. The introduction of the log-link leads naturally to the construction of log- shells at each v ∈ V. If, for simplicity, v ∈ Vbad, then the log-shell at v is given, roughly speaking, by the compact additive module I def −1 · O× ⊆ ⊆ v = pv logv( Kv ) Kv F v [cf. Definition 1.1, (i), (ii); Remark 1.2.2, (i), (ii)]. One has natural functorial algorithms for constructing various versions of the notion of a log-shell — i.e., mono-analytic/holomorphic and ´etale-like/Frobenius-like —fromD-/D- /F -/F-prime-strips [cf.
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