A two term Kuznecov sum formula

Abstract

A period integral is the average of a Laplace eigenfunction over a compact submanifold. Much like for the Weyl law, one can obtain improved estimates on period integrals given geometric or dynamical assumptions on the geodesic flow. While there are many results improving bounds on period integrals, there have been none which improve the remainder of the corresponding sum formula. In this talk, we discuss a recent joint work with Emmett Wyman. We show that an improvement to the remainder term of this sum formula reveals a lower-order oscillating term whose behavior can be described by the dynamics of the geodesic flow. Moreover, this oscillating second term illuminates bounds on period integrals.