Submission date: 8 July 2021

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 many aspects, it aims to mimic o-minimality for non-Archimedean fields. In this article we develop the first Diophantine applications of Hensel minimality. We prove a Pila-Wilkie type of 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. Finally, we give three examples showcasing the necessity of this new dimension counting.

Mathematics Subject Classification: Primary 03C99, 14G05; Secondary 03C65, 11G50, 11D88, 03C98.

Keywords and phrases: Pila-Wilkie theorem, non-Archimedean geometry, tame geometry on henselian valued fields, analogues to o-minimality, model theory of valued fields, rational points of bounded height, parametrizations.

Full text arXiv 2107.03643: pdf, ps.