Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

Abstract

Let $F$ be a totally real number field and $n\ge 3$. Let $\Pi$ and $\pi$ be cuspidal automorphic representations for ${\mathrm{PGL}}_{n+1}(F)$ and ${\mathrm{PGL}}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,\Pi \otimes \tilde{\sigma })$ and $L(1/2,\sigma \otimes \tilde{\pi })$ for certain sequences of $\sigma$ varying over cuspidal automorphic representations for ${\mathrm{PGL}}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,\Pi \otimes \tilde{\sigma })L(1/2,\sigma \otimes \tilde{\pi })$ over a conductor aspect family of $\sigma$.

62 pages, to appear in American Journal of Math

Cited by in corpus (1)

• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications