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.