# Number Theory Seminar

The relative trace formula is a powerful tool to study general automorphic forms by harmonic analysis. Some of its concrete forms can be computed explicitly, e.g., the Kuznetsov formula. However, in general it is difficult to calculate the geometric side of relative trace formulas. So typically people compare the relative trace formulas on two groups by matching their geometric sides and thus obtain relations on representations of these groups. In this talk we will give an introduction to the relative trace formula theory and provide a new application.

The talk will start with the classical Selberg trace formula. Then we will speak the adelic language and introduce the framework of the relative trace formula based on a concrete example: the Jacquet-Rallis trace formula. After the preparation we are ready to talk about a new relative formula which can be seen as a ``section" of the Jacquet-Rallis trace formula. We give an asymptotic computation of the geometric side. Consequently, we obtain interesting information of L-functions on U(3)\times U(2). This is joint work with Philippe Michel and Dinakar Ramakrishnan.