Drinfeld Moduli Schemes and Automorphic Forms

Abstract

We develop Drinfeld’s theory of elliptic modules and their moduli schemes to establish the correspondence of irreducible Galois representations and cuspidal automorphic representations – of GL(r) over a function field – which have a cuspidal local component, on realizing it in the etale cohomology with compact support of the geometric fiber of the moduli scheme. The comparison is based on a comparison of the simple trace formula with Deligne’s conjecture form of the Lefschetz fixed point formula on Q`-adic cohomology with compact support. The Ramanujan conjecture for such cuspidal representations follows, but this we deduce also from congruence relations for Hecke correspondences and the Grothendieck fixed point formula for powers of the Frobenius. The restriction of having a cuspidal local component was removed by L. Lafforgue on developing Drinfeld’s theory of Shtukas.