Cusp forms on GSp(4) with SO(4)-periods

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The SaitoKurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

Original languageEnglish
Pages (from-to)855-919
Number of pages65
JournalInternational Journal of Number Theory
Issue number4
StatePublished - Jun 2011
Externally publishedYes


  • Fourier summation formula
  • Periods of automorphic forms
  • SaitoKurokawa lifting
  • fundamental lemma
  • relative orbital integrals
  • symplectic group


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