Construction of Poincaré-type series by generating kernels

2020/02/20 by Kara, Yasemin, Kumari, Moni, Marzec, Jolanta +3 · 1 citation
#11F72 #35J05 #43A22 #58J35 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2002.09061

Abstract

Let Γ⊂ \textrmPSL₂(\mathbb R) be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let \widetildeΓ be its cover in \textrmSL₂(\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∈\mathbb R under the action of \widetildeΓ. The space of such functions admits the action of the hyperbolic Laplacian Δ_k of weight k. Following an approach of Jorgenson, von Pippich and Smajlović (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.

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