Explicit refinements of B\"ocherer's conjecture for Siegel modular forms of squarefree level

TL;DR

This paper refines B"ocherer's conjecture for degree 2 Siegel modular forms of squarefree level, linking Fourier coefficients with L-values through explicit local integral computations, impacting understanding of their arithmetic properties.

Contribution

It provides an explicit refinement of B"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, with detailed local integral calculations.

Findings

Derived explicit formulas relating Fourier coefficients to L-values.

Identified consequences for the arithmetic properties of Fourier coefficients.

Connected local integral computations to the global Gan-Gross-Prasad conjecture.

Abstract

We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.