Abstract:

The trace formula, invented in the 1950's by Eichler and Selberg, is one of the most powerful tools for studying automorphic forms. It can be viewed as a generalization of the Poisson Summation Formula $$sum_{n=-infty}^{infty} f(n) = sum_{n=-infty}^{infty} hat f(n)$$ Automorphic forms have been studied intensively in recent decades, largely motivated by the deep but still mysterious connections they have withnumber theory. However, one of the difficulties is that in general, automorphic forms are hard (probably impossible) to construct explicitly and we are forced to study them indirectly. The trace formula allows us to do this by computing the trace of certain integral operators acting on spaces of automorphic forms. In this lecture I will explain what the trace formula is, describe some applications, try to mention some more recent extensions (the so-called "relative trace formulas"). No prior knowledge of automorphic forms assumed.