标题： Poincaré级数在平均意义上的非零性 显示英文标题

标题： Non-vanishing of Poincaré Series on Average

摘要： 我们研究同余子群的Poincaré级数何时不恒等于零。 我们证明，当权$k$或指标$m$在二进区间内变化时，具有适当参数的几乎所有Poincaré级数都不会消失。 关键的是，在这些权或指标上“平均”分析这个问题，使我们能够在指标$m$显著大于$k^2$的范围内证明非零性——在这个范围内，证明单个Poincaré级数的非零性仍超出当前方法的范围。 显示英文摘要

摘要： We study when Poincar\'e series for congruence subgroups do not vanish identically. We show that almost all Poincar\'e 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 Poincar\'e series remains out of reach of current methods.