Title: Automorphic forms as rational points

Abstract: In what sense can automorphic forms or Galois representations be viewed as rational points on an algebraic variety? One way to explore this question is by counting arguments. The first result in this direction dates back to an early theorem of Drinfeld, which computes the number of 2-dimensional Galois representations of a function field in positive characteristic; the resulting expression is reminiscent of a Lefschetz fixed point theorem on a smooth algebraic variety over a finite field. More recently it was observed that in the number field setting there are formal similarities between the asymptotic counting problems for rational points on Fano varieties and for automorphic representations on reductive algebraic groups. Very little is known in the latter context. ll discuss joint work on this topic with Djordje Milicevic, in which we solve the automorphic counting problem on the general linear group. Our results can be viewed as being analogous to the well-known result of Schanuel on the number of rational points of bounded height on projective spaces.