Abstract: A classical problem in the theory of automorphic forms is to count the number of eigenvalues of the Laplacian acting on cuspidal functions on the quotient of the upper half plane by a lattice R. For R=SL(2,Z) (or a congruence subgroup thereof) the answer is given by Selberg's Weyl law while for the higher rank situation it was established by Mueller and Lindenstrauss-Venkatesh. Additional to the Laplacian there is another large family of operators, namely the Hecke operators attached to SL(2,Z). Sarnak proved that the eigenvalues of the Hecke operators are on average equidistributed with respect to the Sato-Tate measure on SL(2,C). I want to explain the analogous result for Hecke-Maass cusp forms on SL(n,Z)\SL(n,R)/SO(n). Moreover, we give an effective upper bound for the error term which generalizes the upper bound for the remainder of the Weyl law for SL(n,R)/SO(n) given by Lapid-Mueller. This has applications to the distribution of low-lying zeros of certain families of automorphic L-functions and to the p-adic Ramanujan conjecture on average. This is joint work with Nicolas Templier.
Thu, 31/12/2015 - 14:30 to 15:30
Manchester Building (Hall 2), Hebrew University Jerusalem