Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$.
The Frobenius endomorphism permutes the set of isomorphism classes of irreducible $\ell$-adic local systems ($\ell \neq p$) with a fixed rank on $X$. In 1981, Drinfeld has calculated the number of fixed points of this permutation in the rank 2 case. Curiously, it looks like the number of $\mathbb{F}_q$-points of a variety defined over $\mathbb{F}_q$ whose existence is not at all obvious.
In this talk, we generalize Drinfeld's result to general rank case, which proves some conjectures of Deligne and Kontsevich. Our method is purely automorphic, in fact we do that by using Arthur-Lafforgue's trace formula.
