Reocurrence and weak approximation over geometric global fields

In this article, we prove a Reocurrence Theorem over function fields of curves over $\mathbf{C}(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$ and over finite extensions of the Laurent series field $\mathbf{C}(\text{\hspace{0.17em}⁣}(x,y)\text{\hspace{0.17em}⁣})$. 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 ${\mathrm{S}\mathrm{L}}_n$ and with finite stabilizers is given at the end of the article.