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

9 Mar 2021  ·  Peter Humphries, Jesse Thorner ·

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 prove (i) a strong average form of effective multiplicity one for $\mathrm{GL}_n$; (ii) that given $\pi\in\mathfrak{F}_n$ defined over $\mathbb{Q}$, the convolution $\pi\times\tilde{\pi}$ has a positive level of distribution in the sense of Bombieri-Vinogradov; (iii) that almost all $L(s,\pi\times\pi')\in \mathcal{S}$ have a hybrid-aspect subconvexity bound on $\mathrm{Re}(s)=\frac{1}{2}$; (iv) a hybrid-aspect power-saving upper bound for the variance in the discrepancy of the measures $|\varphi(x+iy)|^2 y^{-2}dxdy$ associated to $\mathrm{GL}_2$ Hecke-Maass newforms $\varphi$ with trivial nebentypus, extending work of Luo and Sarnak for level 1 cusp forms; and (v) a nonsplit analogue of quantum ergodicity: almost all restrictions of Hilbert Hecke-Maass newforms to the modular surface dissipate as their Laplace eigenvalues grow. read more

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Number Theory