Reocurrence and weak approximation over geometric global fields

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.