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On Mochizuki's idea of Anabelomorphy and its applications

Kirti Joshi

TL;DR

This work develops and systematically tests the notion of anabelomorphy, Mochizuki’s idea of changing base fields in anabelian terms, across a wide spectrum of number-theoretic and geometric contexts. By defining amphoricity and its failures, it demonstrates precise invariance and non-invariance phenomena for Galois representations, $p$-adic Hodge theory, local Langlands correspondences, and constructions of varieties, while connecting these to perfectoid spaces and archimedean aspects. The results include synchronization principles for principal series and GL$_n$-representations under anabelomorphy, density theorems relating anabelomorphic connectivity to global geometry, and a suite of examples showing both amphoricity and non-amphoricity of key invariants (e.g., discriminants, Artin/Swan conductors, L-invariants). The work also forges conceptual bridges to Mochizuki’s Ind1 indeterminacy and to broader programmatic themes (arithmetic Teichmüller spaces, higher-dimensional local fields, and differential equations), with concrete open questions highlighted for future study.

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.

On Mochizuki's idea of Anabelomorphy and its applications

TL;DR

This work develops and systematically tests the notion of anabelomorphy, Mochizuki’s idea of changing base fields in anabelian terms, across a wide spectrum of number-theoretic and geometric contexts. By defining amphoricity and its failures, it demonstrates precise invariance and non-invariance phenomena for Galois representations, -adic Hodge theory, local Langlands correspondences, and constructions of varieties, while connecting these to perfectoid spaces and archimedean aspects. The results include synchronization principles for principal series and GL-representations under anabelomorphy, density theorems relating anabelomorphic connectivity to global geometry, and a suite of examples showing both amphoricity and non-amphoricity of key invariants (e.g., discriminants, Artin/Swan conductors, L-invariants). The work also forges conceptual bridges to Mochizuki’s Ind1 indeterminacy and to broader programmatic themes (arithmetic Teichmüller spaces, higher-dimensional local fields, and differential equations), with concrete open questions highlighted for future study.

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.

Paper Structure

This paper contains 79 sections, 74 theorems, 136 equations, 1 table.

Table of Contents

  1. Introduction
  2. What is Anabelomorphy?
  3. Amphoricity and anabelomorphy of $p$-adic fields
  4. Anabelomorphy and Galois representations
  5. Anabelomorphy and the local Langlands Correspondence
  6. Constructions of varieties via anabelomorphy
  7. Local anabelomorphy as Galois Theoretic Surgery on Number Fields
  8. Weak anabelomorphy
  9. Anabelomorphy in perfectoid spaces, the archimedean case, and differential equations
  10. Relationship to Mochizuki's approach and his 'Indeterminacy Ind1'
  11. A picturesque way of thinking about anabelomorphy
  12. Acknowledgments
  13. Anabelomorphy, amphoric quantities and amphoras
  14. Definitions
  15. Anabelomorphy of quasi-projective varieties
  16. ...and 64 more sections

Key Result

Proposition 2.2.3

Suppose $K,L$ are finite fields, $p$-adic fields or number fields. Any anabelomorphism $X/K\stackrel{\alpha}{\leftrightsquigarrow} Y/L$ of geometrically connected, smooth, quasi-projective varieties induces an anabelomorphism

Theorems & Definitions (198)

  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3
  • Remark 2.1.4
  • Definition 2.1.5
  • Remark 2.1.6
  • Definition 2.2.1
  • Remark 2.2.2
  • Proposition 2.2.3
  • proof
  • ...and 188 more