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

Abstract

Let F n be the set of all cuspidal automorphic representations π of GL n with unitary central character over a number field F . We prove the first unconditional zero density estimate for the set S = {L(s, π × π ′ ) : π ∈ F n } of Rankin-Selberg L-functions, where π ′ ∈ F n ′ is fixed. We use this density estimate to prove:(i) a strong average form of effective multiplicity one for GL n ;(ii) that given π ∈ F n defined over Q, the convolution π × π has a positive level of distribution in the sense of Bombieri-Vinogradov; (iii) that almost all L(s, π×π ′ ) ∈ S have a hybrid-aspect subconvexity bound on Re(s) = 1 2 ; (iv) a hybrid-aspect power-saving upper bound for the variance in the discrepancy of the measures |ϕ(x + iy)| 2 y −2 dxdy associated to GL 2 Hecke-Maaß newforms ϕ 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.