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

Binyamini, Gal; Kato, Fumiharu

doi:10.2140/ant.2025.19.1581

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

Gal Binyamini and Fumiharu Kato

Vol. 19 (2025), No. 8, 1581–1619

DOI: 10.2140/ant.2025.19.1581

Abstract

We establish a rigid-analytic analog of the Pila–Wilkie counting theorem, giving subpolynomial upper bounds for the number of rational points in the transcendental part of a $ℚ_{p}$ -analytic set and the number of rational functions in a $𝔽_{q} ((t))$ -analytic set. For $ℤ ((t))$ -analytic sets, we prove such bounds uniformly for the specialization to every nonarchimedean local field.

Keywords

rigid-analytic spaces, Pila–Wilkie theorem, rational points, point counting

Mathematical Subject Classification

Primary: 11G50, 14G05, 14G22

Milestones

Received: 12 July 2023

Revised: 17 May 2024

Accepted: 3 September 2024

Published: 12 June 2025

Authors

Gal Binyamini
Department of Mathematics Weizmann Institute of Science Rehovot Israel

Fumiharu Kato
Department of Mathematics Tokyo Institute of Technology Tokyo Japan

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Vol. 19, No. 8, 2025