On some asymptotic formulas for curves in positive characteristic

Abstract

We prove some general asymptotic formula for the values of $L$-function of a sequence of constructible ${\mathbb{Q}}_l$-sheaves on curves over ${\mathbb{F}}_q$ with some good asymptotic properties. We also give the asymptotic formula for the number of points on the stack ${\mathrm{Bun}}_G$ for asymptotically exact sequence of curves and a split reductive group $G$. In the case of ${\mathrm{GL}}_n$ we prove that the same formula holds if we take only semistable points in the account.