Members’ Seminar

A classical problem in the theory of automorphic forms is to count the number of Laplace eigenfunctions on the quotient of the upper half plane by a lattice $L$. For $L$ a congruence subgroup in $\mathrm{SL}(2,\mathbb Z)$ the Weyl law was proven by Selberg giving an asymptotic count for these eigenfunctions. Further, Sarnak studied the distribution of the Hecke eigenvalues of these eigenfunctions. In higher rank, Lindenstrauss-Venkatesh proved the Weyl law for Hecke-Maass forms on $\mathrm{SL}(2,\mathbb Z) \backslash \mathrm{SL}(n,\mathbb R)/ \mathrm{SO}(n)$. We shall explain how the Hecke eigenvalues of these forms distribute (with an effective error term). An important feature of the proof is the use of non-compactly supported test functions in Arthur's trace formula for $\mathrm{GL}(n)$. This makes it necessary to deal with Arthur's global coefficients, and to prove germ estimates for real orbital integrals over certain unbounded non-continuous functions. Consequences of our result are the $k$-level distribution with restricted support for low-lying zeros of certain families of automorphic L-functions, and a bound towards the Ramanujan conjecture on average. This is joint work with Nicolas Templier.