On some asymptotic formulas for curves in positive characteristic

TL;DR

This paper establishes asymptotic formulas for $L$-functions of sheaves on curves over finite fields and for the point counts on moduli stacks of bundles, with specific results for $ ext{GL}_n$.

Contribution

It provides new asymptotic formulas for $L$-functions and point counts on stacks in positive characteristic, including semistable cases for $ ext{GL}_n$.

Findings

Asymptotic formulas for $L$-functions of sheaves on curves.

Asymptotic point count formulas for $ ext{Bun}_G$ stacks.

Results for semistable points in $ ext{GL}_n$ case.

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.