Title:  Lang-Weil type estimates in finite difference fields

Speaker：Jinhe (Vincent) Ye (University of Oxford)

Time: 2025/01/02 15:00-16:00

Location: Quan29

Zoom room: 717 463 6082

Organizer: Kyle Gannon (BICMR)

Abstract：

　　A difference field is a field equipped with a distinguished automorphism and a difference variety is the natural analogue of an algebraic variety in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields. For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to the Lang-Weil estimate for difference varieties in finite difference fields. The proof uses the model theory of pseudofinite difference fields and increasing valued difference fields. Particularly, some interesting interactions between the nonstandard Frobenius and the non-archimedean topology occur. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.