On Mochizuki's idea of Anabelomorphy and its applications

TL;DR

This paper explores Mochizuki's concept of anabelomorphy, demonstrating its significance in arithmetic geometry by studying its implications in Galois representations, automorphic forms, and introducing the notion of anabelomorphically connected number fields.

Contribution

It formalizes the concept of anabelomorphy, applies it to various arithmetic contexts, and introduces the notion of anabelomorphically connected number fields with new results.

Findings

Establishment of arithmetic consequences of anabelomorphy

Introduction of anabelomorphically connected number fields

Results relating local Galois groups to number field relations

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.