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Topics in Absolute Anabelian Geometry II: Decomposition Groups, manuscript

By Shinichi Mochizuki


� � � 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 a earlier paper of the author concerning the categorytheoretic representation of holomorphic structures via either the topological group SL2 ( ) 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 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 operatio

Year: 2013
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