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

Let Fn be the set of all cuspidal automorphic representations π of GLn 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 ∈ Fn} of Rankin–Selberg L-functions, where π ∈ Fn′ is fixed. We use this density estimate to prove: (i) a strong average form of effective multiplicity one for GLn; (ii) that given π ∈ Fn defined over Q, the convolution π × π̃ 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)|2y−2dxdy associated to GL2 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.