Construction of Poincaré-type series by generating kernels

Abstract

Let Γ ⊂ PSL 2 (R) be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let Γ be its cover in SL 2 (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 ∈ R under the action of Γ. The space of such functions admits the action of the hyperbolic Laplacian ∆ k of weight k. Following an approach of [JvPS16] (where k = 0), we use the spectral expansion associated to ∆ 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 ∆ k .