On the non-vanishing of Hilbert Poincar\'e series
Number Theory
2023-12-08 v1
Abstract
We prove that if has small norm with respect to the level and the weight, the -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 . Finally, we pass to the adelic setting and interpret the problem via Hecke operators.
Keywords
Cite
@article{arxiv.2312.04169,
title = {On the non-vanishing of Hilbert Poincar\'e series},
author = {Mingkuan Zhang and Yichao Zhang},
journal= {arXiv preprint arXiv:2312.04169},
year = {2023}
}