2022/10/20 by Kirti Joshi, Joshi, Kirti · 1 citation
Mathematics · #11G07 #11G20 #11G35 #14G20 #14G40 #14G45 #14G99 #30F60 #32G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.11635
openalex publication_date 2022/10/20 · openalex created_date 2022/10/30 · openalex updated_date 2026/07/28
Let p be a prime number. Let X/E be a geometrically connected, smooth, quasi-projective variety over a finite extension E/ℚp. In this paper I demonstrate the existence of isomorphs of the tempered (and hence also étale) fundamental group of X/E which are labeled by distinct arithmetic holomorphic structures, just as isomorphs of the fundamental group of a Riemann surface Σ may be labeled by Riemann surfaces (i.e. complex holomorphic structures) Σ' in the Teichmuller space of Σ. 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.