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arXiv · 1905.04909

Converse theorems for automorphic distributions and Maass forms of level N

Tadashi Miyazaki·Fumihiro Sato·Kazunari Sugiyama·Takahiko Ueno
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Abstract

We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms studied by Peter and the fourth author.

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Tadashi Miyazaki, Fumihiro Sato, Kazunari Sugiyama, Takahiko Ueno. 2019-05-13. Converse theorems for automorphic distributions and Maass forms of level N. https://arxiv.org/abs/1905.04909

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