p-adic model theory, p-adic integrals, Euler products, and zeta functions of groups

2020/07/17 by Jamshid Derakhshan, Derakhshan, Jamshid
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #math.LO #math.NT

paper · pdf · doi:10.48550/arxiv.2007.09242

arxiv created 2020/07/17 · openalex publication_date 2020/07/17 · arxiv updated 2020/07/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We give a survey of Denef's rationality theorem on p-adic integrals, its uniform in p versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such p-adic integrals over all p, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from definable equivalence relations and p-adic elimination of imginaries, which have applications to counting representations of groups.

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