Gal(Q_p/Q_p) as a geometric fundamental group

2014/04/28 by Jared Weinstein, Weinstein, Jared
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1404.7192

22 pages

arxiv created 2014/04/28 · openalex publication_date 2014/04/28 · arxiv updated 2014/04/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of Q_p is the étale fundamental group of a certain object Z which is defined over an algebraically closed field. Thus, local Galois representations correspond to local systems on Z. In brief, Z is a (non-representable) quotient of a perfectoid space. The construction combines two themes: the fundamental curve of p-adic Hodge theory (due to Fargues-Fontaine) and the tilting equivalence (due to Scholze).

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