Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Martin Dickson, Ameya Pitale, Abhishek Saha, Ralf Schmidt

Published 2015-12-22Version 1

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. We also compute and write down the relevant $\varepsilon$-factors for all dihedral twists of non-supercuspidal representations of $GSp_4$ over a non-archimedean local field. By comparing this with the conditions for a local Bessel model to exist, we verify the local Gan-Gross-Prasad conjecture in these cases for the generic L-packets and also demonstrate its severe failure for the non-generic L-packets.