Non-vanishing of Poincaré Series on Average

TL;DR

This paper addresses the non-vanishing problem for Poincaré series $P_{k,m,q,\chi}$ on average, showing that almost all such series are nonzero when averaged over weight $k$ or index $m$ within dyadic ranges. The authors reduce the problem to bounding second moments of the squared Petersson norms, $\|\widetilde{P}_{k,m,q,\chi}\|_2^2-1$, which are expressed in terms of twisted Kloosterman sums and Bessel functions via $\Delta_{k,q,\chi}(m,m)$. They establish two main second-moment bounds: one averaged over $k$ (weight-average) and another over $m$ (index-average), each incorporating the Weil-type bounds for Kloosterman sums, Neumann's addition theorem for Bessel sums, and smoothing techniques. From these bounds, they prove that, in specified ranges (e.g., $m(m,q) \le K^{3-\!200\varepsilon} q^{2}/g(q')^{2}$ for the weight-average and $M \le k^{1000}$ for the index-average), almost all Poincaré series in the respective dyadic intervals do not vanish identically, improving prior results and extending non-vanishing into regimes with $m$ significantly larger than $k^{2}$. This has implications for understanding the distribution of nonzero Poincaré series in families of modular forms and for related spectral questions on congruence subgroups.

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.

Theorems & Definitions (36)

• Theorem 1.1
• Remark
• Theorem 1.2
• Remark
• Lemma 2.1: Bessel Function Bounds
• proof
• Lemma 2.2: Neumann's addition theorem
• proof
• Lemma 2.3
• proof