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AN EQUIDISTRIBUTION THEOREM FOR HOLOMORPHIC SIEGEL MODULAR FORMS FOR $\mathit{GSp}_{4}$ AND ITS APPLICATIONS

Part of: Algebraic number theory: global fields Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  22 February 2018

Henry H. Kim ,
Satoshi Wakatsuki  and
Takuya Yamauchi
Henry H. Kim
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada (henrykim@math.toronto.edu) Korea Institute for Advanced Study, Seoul, Korea
Satoshi Wakatsuki
Affiliation:
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 920-1192, Japan (wakatsuk@staff.kanazawa-u.ac.jp)
Takuya Yamauchi
Affiliation:
Mathematical Inst. Tohoku Univ. 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, Japan (tyamauchi@m.tohoku.ac.jp)
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Abstract

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $\mathit{GSp}_{4}/\mathbb{Q}$ 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 $L$-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 $A,B_{1}$ in Theorem 1.1 which have not been studied and a mysterious second term $B_{2}$ 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 $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

Keywords

trace formulaHecke operatorsSiegel modular forms

MSC classification

Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 11R45: Density theorems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 351 - 419
Copyright
© Cambridge University Press 2018

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Footnotes

The first author is partially supported by NSERC. The second author is partially supported by JSPS Grant-in-Aid for Scientific Research (nos. 26800006, 25247001, 15K04795). The third author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) no. 15K04787.

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