On Mochizuki's idea of Anabelomorphy and its applications

Abstract

I coined the term anabelomorphy (pronounced as anabel-o-morphy) as a concise way of expressing Mochizuki's idea of "anabelian way of changing ground field, rings etc." which was he has introduced in his work on his Inter-Universal Teichmuller Theory. This paper demonstrates the usefulness of this idea by studying its ramifications in the more familiar arithmetic contexts such as the theory of Galois representations, automorphic forms and related areas and establish a number of results which are of independent arithmetic interest. I also introduce the notion of anabelomorphically connected number fields in which two number fields are related by the existence of topological isomorphism between the local Galois groups at a finite list of primes of both the number fields and prove some results illustrating arithmetic consequences of this notion. The Introduction provides a detailed discussion and summary of all the results proved in this paper.

Changes: 77 pages. Referee recommended changes and cleanup and addition of two Subsections 7.7, 14.1; moved previous Subsection 1.10 to Appendix. Previous Changes: 74 Pages. Revision, reorganization (proofs are cleaned up and trimmed), some additions (e.g. §6.7, and some new questions included)