Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- 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.