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.