Zero density estimates for $\mathrm{GL}_2$ automorphic $L$-functions

Abstract

We prove zero density estimates for $L$-functions of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$. We show that $N_\pi(\sigma, T) \ll T^{\frac{5}{2}(1 - \sigma) + o(1)}$, where $N_\pi(\sigma, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $L(s,\pi)$ with $\beta \geq \sigma$ and $|\gamma| \leq T$. The key input is an extension of the Guth$\unicode{x2013}$Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker $\ell^q$ bound.