ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)
PI : Michael HARRIS
The goal of this talk is to explain the proof of the following result: Let X be a curve over a finite field, and let $\sigma: \pi_1(X)\to G^L$ be an irreducible unramified Galois representation. Assume that the restriction of $\sigma$ to the geometric fundamental group of $X$ maps to the Cartan subgroup $T^L\subset G^L$. Then to $\sigma$ there corresponds an unramified automorphic function on $G$. The idea of the construction of the following: we will first construct the corresponding automorphic sheaf over the algebraic closure of our ground field. This will be done using the theory of geometric Eisenstein series. We will then use the geometric functional equation to endow it with the structure of Weil sheaf. The sought-for function will be obtained by taking traces of the Frobenius.