Asymptotics of traces of Hecke operators and Arthur's trace formula

We will discuss the spectral theory of the locally symmetric varieties associated to reductive groups defined over number fields. A fundamental tool is Arthur's trace formula, which in a first approximation expresses the duality between the spectrum and the conjugacy classes of the group of rational points. However, the picture is complicated by the contribution of the continuous spectrum to the spectral side and by the terms corresponding to non-elliptic conjugacy classes on the geometric side. We will review the absolute convergence and continuity of the trace formula for a large natural class of test functions. We will then discuss a concrete application to the asymptotics of traces of Hecke operators (for classical groups and G2). This is joint work with Erez Lapid, Werner Mueller and Jasmin Matz.