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 $\beta \ge \sigma$ and $|\gamma |\le T$. The key input is an extension of the Guth$\unicode{x2013}$Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker ${\ell }^q$ bound.

arXiv comment36 pages