An equidistribution theorem for holomorphic siegel modular forms for GSP4 and its applications

Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi

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4 Citations (Scopus)


We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for in various aspects. A main tool is Arthur's invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83-120] and Shin-Templier [Sato-Tate theorem for families and low-lying zeros of automorphic -functions, Invent. Math.203(1) (2016) 1-177] used Euler-Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms in Theorem 1.1 which have not been studied and a mysterious second term also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato-Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor -functions and degree 5 standard -functions of holomorphic Siegel cusp forms.

Original languageEnglish
Pages (from-to)351-419
Number of pages69
JournalJournal of the Institute of Mathematics of Jussieu
Issue number2
Publication statusPublished - 2020 Mar 1


  • Hecke operators
  • Siegel modular forms
  • trace formula


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