Construction of Poincaré-type series by generating kernels

Abstract

Let $\Gamma \subset {\text{PSL}}_2(\mathbb{R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\tilde{\Gamma }$ be its cover in ${\text{SL}}_2(\mathbb{R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in \mathbb{R}$ under the action of $\tilde{\Gamma }$. The space of such functions admits the action of the hyperbolic Laplacian ${\Delta }_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlović (where $k=0$), we use the spectral expansion associated to ${\Delta }_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of ${\Delta }_k$.

Submitted to the Proceedings of the Women in Numbers Europe 3 Conference