Topics in Absolute Anabelian Geometry III: Global Reconsruction Algorithms

J. Math. Sci. Univ. Tokyo
Vol. 22 (2015), No. 4, Page 939–1156.

Mochizuki, Shinichi
Topics in Absolute Anabelian Geometry III: Global Reconsruction Algorithms
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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 cuspi- dalization 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 topo- logical group SL2(R) or the use of “parallelograms, rectangles, and squares”, to develop a certain global formalism for certain hyperbolic orbicurves related to a once-punctured 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 isomor- phism”bothinthesensethatitfailstorespecttheringstructures involved and in the sense (cf. Frobenius morphisms in positive char- acteristic!)thatithastheeffectofexhibitingthe“mass”represented by its domain as a “somewhat smaller collection of mass” than the “mass”representedbyitscodomain,thisglobalformalismallowsone 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 rem- iniscent of the isomorphism induced on differentials by a Frobenius lifting, once one divides by p. 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 Teichmu ̈ller theory.

Keywords: Absolute anabelian geometry, mono-anabelian, core, Belyi cuspidalization, elliptic cuspidalization, arithmetic holomorphic structure, mono-analytic, log-Frobenius, log-shell, log-volume.

Received: 2008-03-27