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

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 ${\mathbb{Q}}_p$-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 nonarchimedean local field.