Simple zeros of $\mathrm{GL}(2)$ $L$-functions

TL;DR

This paper establishes a lower bound on the number of simple zeros of the completed $L$-function associated with primitive holomorphic forms of arbitrary level, advancing understanding of zero distributions for these functions.

Contribution

It provides the first power bound for simple zeros of $L$-functions of non-trivial level, improving previous results that were logarithmic or infinite in nature.

Findings

Proves $ ext{Omega}(T^ ext{delta})$ simple zeros with $ ext{delta}<2/27$ for $L$-functions of forms with arbitrary level.

Improves bounds on the number of simple zeros for trivial level ($N=1$) forms.

First power bound result for non-trivial level forms in this context.

Abstract

Let $f \in S_k(\Gamma_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $\Omega\left(T^\delta\right)$ simple zeros with imaginary part in $\left[-T, T\right]$, for any $\delta < \frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $\Omega(\log\log\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros.