An equidistribution theorem for holomorphic Siegel modular forms for $GSp_4$

TL;DR

This paper establishes an equidistribution theorem for holomorphic Siegel cusp forms for GSp_4 over Q, utilizing Arthur's trace formula with novel geometric contributions, and explores applications like Sato-Tate and L-function zeros.

Contribution

It introduces a new approach using a pseudo-coefficient of a holomorphic discrete series in Arthur's trace formula, revealing new second main terms and broadening the analysis of weights.

Findings

Proves an equidistribution theorem for GSp_4 holomorphic Siegel cusp forms.

Identifies new geometric contributions and second main terms in the trace formula.

Provides applications to Sato-Tate, Hecke fields, and zeros of L-functions.

Abstract

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincar\'e 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 the main theorem 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.