Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- Arithmetic Holomorphic Structures

TL;DR

This paper constructs labeled isomorphs of the fundamental group of algebraic varieties over p-adic fields, paralleling complex holomorphic structures, and discusses their implications in arithmetic geometry and Mochizuki's work.

Contribution

It demonstrates the existence of arithmetic holomorphic structures that label isomorphs of fundamental groups of varieties over p-adic fields, extending the analogy with complex structures.

Findings

Existence of labeled isomorphs of fundamental groups over p-adic fields.

Introduction of arithmetic holomorphic structures in this context.

Connection to Mochizuki's arithmetic holomorphic structures.

Abstract

Let $p$ be a prime number. Let $X/E$ be a geometrically connected, smooth, quasi-projective variety over a finite extension $E/\mathbb{Q}_p$. In this paper I demonstrate the existence of isomorphs of the tempered (and hence also \'etale) fundamental group of $X/E$ which are labeled by distinct arithmetic holomorphic structures, just as isomorphs of the fundamental group of a Riemann surface $\Sigma$ may be labeled by Riemann surfaces (i.e. complex holomorphic structures) $\Sigma'$ in the Teichmuller space of $\Sigma$. This is the starting point of the theory elaborated in [Joshi, 2021a,b,c, 2022] for which this paper is intended as an brief sketch and announcement. Arithmetic holomorphic structures introduced here also provide distinct arithmetic holomorphic structures used by Shinichi Mochizuki in [Mochizuki,2021a,b,c,d]. Since the question of whether or not there exists distinct arith. hol. structures in [Mochizuki,2021a,b,c,d] was raised in [Scholze and Stix], I include a discussion of [Scholze and Stix]. See the introduction for additional details.