Abstract: In this talk we focus on the following three automorphic forms on a Fuchsian group of the first kind with at least one cusp: the Eisenstein series and the Niebur-Poincaré series associated to the cusp at infinity, and the resolvent kernel/Green's function. We discuss how those functions can be viewed as building blocks for describing log-norms of some meromorphic functions in terms of their divisors and derive a generalization of the Rorlich-Jensen type formula, which is based on an evaluation of the Petersson inner product of the Niebur-Poincaré series with the suitably regularized Green's function. Then, we turn our attention to the generating functions of the Niebur-Poincaré series and its derivative at s=1. Both functions depend upon two variables in the upper half-plane. We prove that, for any Fuchsian group of the first kind, the generating function of the Niebur-Poincaré series in each variable is a polar harmonic Maass form of a certain weight, describe its polar part and discuss how it can be viewed as a building block for describing weight two meromorphic modular forms in terms of their divisors. Moreover, we prove that the generating function of the derivative of the Niebur-Poincaré series at s=1 can be expressed, up to a certain function appearing in the Kronecker limit formula, as a derivative of an automorphic kernel associated to a new point-pair invariant expressed in terms of the Rogers dilogarithm.

The talk is based on the joint work with Kathrin Bringmann, James Cogdell and Jay Jorgenson.