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

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 $\mathcal{S}=\{L(s,\pi \times {\pi }^{\prime }):\pi \in {\mathfrak{F}}_n\}$ of Rankin-Selberg $L$-functions, where ${\pi }^{\prime }\in {\mathfrak{F}}_{n^{\prime }}$ is fixed. We use this density estimate to establish (i) a hybrid-aspect subconvexity bound at $s=\frac{1}{2}$ for almost all $L(s,\pi \times {\pi }^{\prime })\in \mathcal{S}$, (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 $\pi \in {\mathfrak{F}}_n$.

32 pages. Significant revision based on referee comments

References in corpus (3)

• Upper bounds on L-functions at the edge of the critical strip
• A Bombieri-Vinogradov theorem for higher rank groups
• The second moment of $\mathrm{GL}(n)\times \mathrm{GL}(n)$ Rankin--Selberg $L$-functions

Cited by in corpus (2)

• A new zero-free region for Rankin-Selberg $L$-functions
• New variants of arithmetic quantum ergodicity