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

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.