Zeros of Rankin-Selberg $L$-functions in families

TL;DR

This paper establishes the first unconditional zero density estimate for families of Rankin-Selberg $L$-functions, leading to new bounds and distribution results for automorphic representations over number fields.

Contribution

It provides the first unconditional zero density estimate for Rankin-Selberg $L$-functions in families, enabling new subconvexity bounds and distribution results for automorphic forms.

Findings

Unconditional zero density estimate for Rankin-Selberg $L$-functions.

Hybrid subconvexity bounds at $s=1/2$ for almost all $L$-functions.

Positive level of distribution in the sense of Bombieri-Vinogradov.

Abstract

Let $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ with unitary central character over a number field $F$. We prove the first unconditional zero density estimate for the set $\mathcal{S}=\{L(s,\pi\times\pi')\colon\pi\in\mathfrak{F}_n\}$ of Rankin-Selberg $L$-functions, where $\pi'\in\mathfrak{F}_{n'}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,\pi\times\pi')\in \mathcal{S}$, (ii) a strong on-average form of effective multiplicity one for almost all $\pi\in\mathfrak{F}_n$, and (iii) a positive level of distribution for $L(s,\pi\times\tilde{\pi})$, in the sense of Bombieri-Vinogradov, for each $\pi\in\mathfrak{F}_n$.