A spectral reciprocity formula and non-vanishing for L-functions on GL(4)×GL(2)

Valentin Blomer, Xiaoqing Li, Stephen D. Miller

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We introduce a new type of summation formula for central values of GL(4)×GL(2) L-functions, when varied over Maaß forms. By rewriting such a sum in terms of GL(4)×GL(1) L-functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL(4,Z), there exist infinitely many Maaß forms π for SL(2,Z) such that L(1/2,Π×π)≠0.

Original languageEnglish (US)
Pages (from-to)1-43
Number of pages43
JournalJournal of Number Theory
StatePublished - Dec 2019

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


  • Kuznetsov formula
  • Moments
  • Non-vanishing
  • Rankin-Selberg L-functions
  • Spectral summation
  • Voronoi summation

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