On higher regulators of Siegel threefolds II: the connection to the special value

Abstract

We establish a connection between motivic cohomology classes over the Siegel threefold and non-critical special values of the degree-four$L$-function of some cuspidal automorphic representations of$\text{GSp}(4)$. Our computation relies on our previous work [On higher regulators of Siegel threefolds I: the vanishing on the boundary, Asian J. Math.19(2015), 83–120] and on an integral representation of the$L$-function due to Piatetski-Shapiro.