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

TL;DR

This paper surveys key results in $p$-adic model theory and integrals, highlighting their applications to counting subgroups, conjugacy classes, and representations in algebraic groups and nilpotent groups.

Contribution

It consolidates known theorems on $p$-adic integrals, explores their uniformity across primes, and discusses applications to various counting problems in group theory and number theory.

Findings

Rationality of $p$-adic integrals established by Denef.

Uniform versions of $p$-adic integrals across all primes.

Applications to counting subgroups, conjugacy classes, and representations.

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.