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.

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.

## Date:

Thu, 31/12/2015 - 14:30 to 15:30

## Location:

Manchester Building (Hall 2), Hebrew University Jerusalem