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.