April 6, 2000
Andre Reznikov
Title: Automorphic functions and representation theory
joint with J. Bernstein, Tel-Aviv. We introduced a new approach into theory of automorphic representations based on a quantitative version of Frobenius reciprocity we proved. Our main tools are natural norms arising on representations. The main goal of our approach is to obtain analytic information on various objects related to automorphic representations. In particular we obtained new bounds on some invariant functionals, Fourier coefficients and $L$-functions of automorphic representations for the group $GL(2)$. As a result we obtained new results towards Ramanujan-Peterson conjecture about the size of Fourier coefficients of cusp forms. The proof is more elementary and valued in greater generality than other approaches. In particular we obtained results for non-arithmetic lattices which are not accessible in other approaches. Similarly we where able to prove non-vanishing of various $GL(2)$ type $L$-functions. In this talk I will explain basic definitions and give a sample of results and conjectures.