Weyl's law with remainder and Hecke operators

Weyl's law describes the asymptotic behavior of the counting function for the discrete or cuspidal spectrum of the Laplace operator on a Riemannian manifold. In this talk, we consider locally symmetric manifolds, which may be compact or non-compact. The Weyl asymptotic is well known, even with an explicit power saving in the remainder term, in the case of compact locally symmetric manifolds and in many other cases. In the case of non-compact manifolds, we only expect Weyl's law to hold in the arithmetic case, and Lindenstrauss-Venkatesh (2007) indeed used Hecke operators (which only exist for arithmetic groups) to establish a general version of the law for the cuspidal spectrum. They did not provide any additional information on the magnitude of the remainder term. We show how to use Hecke operators and Arthur's trace formula to prove Weyl's law for the cuspidal spectrum with a power saving in the remainder term. This is joint work with Erez Lapid and Jasmin Matz.