ON HIGHER REGULATORS OF SIEGEL THREEFOLDS I: THE VANISHING ON THE BOUNDARY

In this paper, we prove the vanishing of a map between the absolute Hodge cohomology spaces of the boundaries of the Baily-Borel compactifications of the product of two modular curves on one side and of the Siegel threefold on the other side. As an application, we construct some 1-extensions of mixed Hodge structures between the trivial Hodge structure and the middle-degree interior cohomology of the Siegel threefold, which come from motivic cohomology. These are conjecturally related to non-critical values of the degree 4 L-function of some cuspidal automorphic representations of GSp(4).