Rational points of rigid-analytic sets: a Pila-Wilkie type theorem

TL;DR

This paper proves a Pila-Wilkie type theorem for rigid-analytic sets over non-archimedean fields, providing bounds on rational points in transcendental parts, extending counting results to p-adic and function field contexts.

Contribution

It establishes a rigid-analytic analog of the Pila-Wilkie theorem, offering uniform bounds for rational points in non-archimedean analytic sets across various fields.

Findings

Sub-polynomial bounds for rational points in p-adic analytic sets

Uniform bounds for specializations over non-archimedean fields

Extension of Pila-Wilkie theorem to rigid-analytic geometry

Abstract

We establish a rigid-analytic analog of the Pila-Wilkie counting theorem, giving sub-polynomial upper bounds for the number of rational points in the transcendental part of a $\mathbb{Q}_p$-analytic set, and the number of rational functions in a $\mathbb{F}_q((t))$-analytic set. For $\mathbb{Z}[[t]]$-analytic sets we prove such bounds uniformly for the specialization to every non-archimedean local field.