Construction of Poincaré-type Series by Generating Kernels

Abstract

Let $\Gamma \subset \operatorname {PSL}_2(\mathbb {R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetilde \Gamma$ be its cover in $\operatorname {SL}_2(\mathbb {R})$. 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 $\widetilde \Gamma$. The space of such functions admits the action of the hyperbolic Laplacian Δ_k of weight k. Following an approach of by Jorgenson et al. (Adv Math 288:887–921, 2016) (where k = 0), we use the spectral expansion associated to Δ_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 Δ_k.