Non-vanishing of automorphic L-functions of prime power level

TL;DR

This paper extends non-vanishing results of automorphic L-functions at the critical point from square-free levels to prime power levels, showing that at least 25% of such L-values do not vanish for large levels.

Contribution

It generalizes previous non-vanishing results to automorphic forms with prime power level, broadening understanding of L-function behavior.

Findings

At least 25% of L-values do not vanish at the critical point for prime power levels.

The result applies to automorphic forms of fixed weight and large prime power level.

Extension of non-vanishing results from square-free to prime power levels.

Abstract

Iwaniec and Sarnak showed that at the minimum 25% of L-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when N is square-free and $\phi(N)\sim N$. In this paper we extend the given result to the case of prime power level $N=p^{\nu}$, $\nu\geq 2$.