TY - JOUR

T1 - An equidistribution theorem for holomorphic siegel modular forms for GSP4 and its applications

AU - Kim, Henry H.

AU - Wakatsuki, Satoshi

AU - Yamauchi, Takuya

N1 - Funding Information:
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.
Publisher Copyright:
© Cambridge University Press 2018.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

KW - Hecke operators

KW - Siegel modular forms

KW - trace formula

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U2 - 10.1017/S147474801800004X

DO - 10.1017/S147474801800004X

M3 - Article

AN - SCOPUS:85042364017

VL - 19

SP - 351

EP - 419

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 2

ER -