### [CANCELLED] Construction of Poincaré-type series by generating kernels

Linfoot Number Theory Seminar

18th March 2020, 11:00 am – 12:00 pm

Fry Building, 2.04

Let $\Gamma\subset\text{\rm PSL}_2(\mathbb{R})$ be a Fuchsian group of the first kind whose fundamental domain $\Gamma\backslash\mathbb{H}$ is of finite volume, and let $\widetilde\Gamma$ be its cover in $\SL_2(\mathbb{R})$. Consider the space of twice continuously differentiable, square-integrable functions on $\mathbb{H}$, 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 action of the hyperbolic Laplacian $\Delta_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlovi\'c (where $k=0$), we use spectral expansion associated to $\Delta_k$ to construct wave distribution and then identify conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincar\'e-type series. As we will show, one of advantages 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 $\Delta_k$. This is joint work with Y. Kara, M. Kumari, K. Maurischat, A. Mocanu and L. Smajlovi\'c.

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