A Pila--Wilkie theorem for Hensel minimal curves

Abstract: Recently, a new axiomatic framework for tameness in henselian valued fields was developed by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen and termed Hensel minimality. In this article we develop Diophantine applications of Hensel minimality. We prove a Pila--Wilkie type theorem for transcendental curves definable in Hensel minimal structures. In order to do so, we introduce a new notion of point counting in this context related to dimension counting over the residue field. We examine multiple classes of examples, showcasing the need for this new dimension counting and prove that our bounds are optimal.