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Zeros of Rankin–Selberg L-functions in families
Published online by Cambridge University Press: 03 April 2024
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
$L(s,\pi \times \widetilde {\pi })$, in the sense of Bombieri–Vinogradov, for each
$\pi \in \mathfrak {F}_n$.
- Type
- Research Article
- Information
- Copyright
- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
References
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