Reocurrence and weak approximation over geometric global fields

TL;DR

This paper proves a recurrence theorem over certain function fields, offering a partial alternative to Chebotarev's theorem, and applies it to weak approximation problems for specific algebraic homogeneous spaces.

Contribution

It introduces a recurrence theorem over geometric global fields, extending the understanding of rational points and weak approximation in these contexts.

Findings

Proves a recurrence theorem over function fields of curves over complex Laurent series.

Provides a partial replacement for Chebotarev's theorem in these fields.

Applies the theorem to weak approximation for homogeneous spaces under SL_n with finite stabilizers.

Abstract

In this article, we prove a Reocurrence Theorem over function fields of curves over $\mathbf{C}(\! (t)\! )$ and over finite extensions of the Laurent series field $\mathbf{C}(\! (x,y)\! )$. This provides a partial replacement to Chebotarev's Theorem over such fields. A concrete application to the study of weak approximation for homogeneous spaces under $\mathrm{SL}_n$ and with finite stabilizers is given at the end of the article.