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arXiv · 2312.04169

On the non-vanishing of Hilbert Poincar\'e series

Mingkuan Zhang·Yichao Zhang
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Abstract

We prove that if $\nu$ has small norm with respect to the level and the weight, the $\nu$-th Hilbert Poincar\'e series does not vanish identically. We also prove Selberg's identity on Kloosterman sums in the case of number fields, which implies certain vanishing and non-vanishing relations of Hilbert Poincar\'e series when the narrow class number is $1$. Finally, we pass to the adelic setting and interpret the problem via Hecke operators.

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Mingkuan Zhang, Yichao Zhang. 2023-12-07. On the non-vanishing of Hilbert Poincar\'e series. https://arxiv.org/abs/2312.04169

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