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 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.