Year: 2025
Type: preprint
Abstract: We study when Poincaré series for congruence subgroups do not vanish identically. We show that almost all Poincaré series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval. Crucially, analyzing the problem `on average' over these weights or indices allows us to prove non-vanishing in ranges where the index $m$ is significantly larger than $k^2$ - a range in which proving non-vanishing for individual Poincaré series remains out of reach of current methods.
Source: arXiv (Cornell University)
Authors: Carmichael, Ned, Kimmel, Noam
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077974312
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