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

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.