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.

中文翻译：

---

全晶西格尔模形式的均分定理及其应用

我们证明了一个全纯 Siegel 尖顶形式族的等分布定理$\mathit{GSp}_{4}/\mathbb{Q}$在各个方面。一个主要工具是亚瑟的不变迹公式。而 Shin [Automorphic Plancherel 密度定理，以色列 J. 数学。192(1) (2012), 83-120] 和 Shin-Templier [Sato-Tate 定理，用于家庭和自守的低位零点$L$-职能，发明。数学。203(1) (2016) 1-177] 在公式中使用无穷远处的 Euler-Poincaré 函数，我们使用全纯离散级数的伪系数来提取全纯 Siegel 尖点形式。然后非半简单贡献来自几何方面，这提供了新的第二个主要术语$A,B_{1}$在尚未研究的定理 1.1 和一个神秘的第二项$B_{2}$也出现在来自半简单元素的第二个主要术语中。此外，我们的明确研究使我们能够处理权重中更一般的方面。我们还给出了几个应用，包括垂直 Sato-Tate 定理、Hecke 场的无界性和 4 次旋量的低位零点$L$- 功能和 5 级标准$L$-全纯西格尔尖点形式的函数。