Spectral reciprocity for GL( n ) and simultaneous non-vanishing of central L -values
Abstract
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 $L(1/2,\Pi\otimes\widetilde{\sigma})$ and $L(1/2,\sigma\otimes\widetilde{\pi})$ 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 $L(1/2,\Pi\otimes\widetilde{\sigma})L(1/2,\sigma\otimes\widetilde{\pi})$ over a conductor aspect family of $\sigma$.
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Algebra and Geometry
Finite Group Theory Research
Publication details
- Journal / source
- American Journal of Mathematics
- Publisher
- Project MUSE
- Volume
- 148
- Issue
- 1
- Pages
- 247–311
- Publication date
- 23 January 2026
- Language
- EN