Lang-Weil type bounds in finite difference fields

Basic data for this talk

Type of talk: scientific talk
Name of speakers: Hils, Martin
Date of talk: 30/05/2023
Talk language: English

Information about the event

Name of the event: Model theory of valued fields
Event location: CIRM (Marseille)

Abstract

(joint work with Ehud Hrushovski, Jinhe Ye and Tingxiang Zou) We prove Lang-Weil type bounds for the number of rational points of difference varieties over finite difference fields, in terms of the transformal dimension of the variety and assuming the existence of a smooth rational point. It follows that in (certain) non-principle ultraproducts of finite difference fields the course dimension of a quantifier free type equals its transformal transcendence degree. The proof uses a strong form of the Lang-Weil estimates and, as key ingredient to obtain equidimensional Frobenius specializations, the recent work of Dor and Hrushovski on the non-standard Frobenius acting on an algebraically closed non-trivially valued field, in particular the pure stable embeddedness of the residue difference field in this context.

Keywords: Lang-Weil type bounds; finite difference fields; model theory