Seminar: Lejla Smajlovic - On some nonholomorphic automorphic forms, their inner products and generating functions (Dec 17, 2025)

Speaker: Lejla Smajlovic (University of Sarajevo)

Title: On some nonholomorphic automorphic forms, their inner products and generating functions

Time: December 17, Wednesday, 2025 at 17:00 (Istanbul LT), 17:30 (Tehran LT), 19:30 (Allahabad LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

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, the Niebur–Poincaré series associated to the cusp at infinity, and the resolvent kernel/Green's function. We discuss how these functions can be viewed as building blocks for describing log-norms of certain meromorphic functions in terms of their divisors and derive a generalization of a Rorlich–Jensen type formula, based on an evaluation of the Petersson inner product of the Niebur–Poincaré series with a suitably regularized Green's function. We then turn our attention to the generating functions of the Niebur–Poincaré series and its derivative at s=1. Both functions depend on 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 explain how it serves as a building block for describing weight two meromorphic modular forms in terms of their divisors. Moreover, we show 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 with a new point-pair invariant expressed in terms of the Rogers dilogarithm. This talk is based on joint work with Kathrin Bringmann, James Cogdell, and Jay Jorgenson.