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102: １３２人目の素数さん [] 2021/04/17(土) 12:52:51.59 ID:cr30r3uy

メモ
https://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20III.pdf
TOPICS IN ABSOLUTE ANABELIAN GEOMETRY III:
GLOBAL RECONSTRUCTION ALGORITHMS
Shinichi Mochizuki
November 2015

Abstract. In the present paper, which forms the third part of a three-part series
on an algorithmic approach to absolute anabelian geometry, we apply the absolute anabelian technique of Belyi cuspidalization developed in the second part,
together with certain ideas contained in an earlier paper of the author concerning the
category-theoretic representation of holomorphic structures via either the topological group SL2(R) or the use of “parallelograms, rectangles, and squares”, to develop
a certain global formalism for certain hyperbolic orbicurves related to a oncepunctured elliptic curve over a number field. This formalism allows one to construct
certain canonical rigid integral structures, which we refer to as log-shells, that
are obtained by applying the logarithm at various primes of a number field. Moreover, although each of these local logarithms is “far from being an isomorphism” both
in the sense that it fails to respect the ring structures involved and in the sense [cf.
Frobenius morphisms in positive characteristic!] that it has the effect of exhibiting
the “mass” represented by its domain as a “somewhat smaller collection of mass”
than the “mass” represented by its codomain, this global formalism allows one to
treat the logarithm operation as a global operation on a number field which satisfies
the property of being an “isomomorphism up to an appropriate renormalization operation”, in a fashion that is reminiscent of the isomorphism induced
on differentials by a Frobenius lifting, once one divides by p.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1606813903/102

103: １３２人目の素数さん [] 2021/04/17(土) 12:53:20.91 ID:cr30r3uy

>>102
つづき

More generally, if one
thinks of number fields as corresponding to positive characteristic hyperbolic curves
and of once-punctured elliptic curves on a number field as corresponding to nilpotent
ordinary indigenous bundles on a positive characteristic hyperbolic curve, then many
aspects of the theory developed in the present paper are reminiscent of [the positive
characteristic portion of] p-adic Teichm¨uller theory.

Contents:
Introduction
§0. Notations and Conventions
§1. Galois-theoretic Reconstruction Algorithms
§2. Archimedean Reconstruction Algorithms
§3. Nonarchimedean Log-Frobenius Compatibility
§4. Archimedean Log-Frobenius Compatibility
§5. Global Log-Frobenius Compatibility
Appendix: Complements on Complex Multiplication

Introduction
§I1. Summary of Main Results
§I2. Fundamental Naive Questions Concerning Anabelian Geometry
§I3. Dismantling the Two Combinatorial Dimensions of a Ring
§I4. Mono-anabelian Log-Frobenius Compatibility
§I5. Analogy with p-adic Teichm¨uller Theory
Acknowledgements
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1606813903/103

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