The heat kernel, theta inversion and zetas on τc\G\K

Abstract

Direct and precise connections between zeta functions with functional equations and theta functions with inversion formulas can be made using various integral transforms, namely Laplace, Gauss, and Mellin transforms as well as their inversions. In this article, we will describe how one can initiate the process of constructing geometrically defined zeta functions by beginning inversion formulas which come from heat kernels. We state conjectured spectral expansions for the heat kernel, based on the so-called heat Eisenstein series defined in [JoL 04]. We speculate further, in vague terms, the goal of constructing a type of ladder of zeta functions and describe similar features from elsewhere in mathematics.