2025/08/24 by Carmichael, Ned, Kimmel, Noam
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.17242
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 k2 - a range in which proving non-vanishing for individual Poincaré series remains out of reach of current methods.