We formulate an explicit refinement of Böcherer'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 Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.
The third author acknowledges the support of the EPSRC grant EP/L025515/1.
"Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level." J. Math. Soc. Japan 72 (1) 251 - 301, January, 2020. https://doi.org/10.2969/jmsj/78657865