On the residue of Eisenstein classes of Siegel varieties

TL;DR

This paper investigates the residues of Eisenstein classes in the motivic cohomology of Siegel varieties, demonstrating their non-triviality on zero-dimensional strata and implications for higher regulator maps.

Contribution

It proves the non-trivial residue of Eisenstein classes on Siegel varieties of any genus by reducing to the Hilbert-Blumenthal case, revealing new insights into their motivic properties.

Findings

Residues of Eisenstein classes are non-trivial on zero-dimensional strata.

Non-vanishing of higher regulator maps is established.

Reduction to Hilbert-Blumenthal case is a key technique.

Abstract

Eisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert-Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of arbitrary genus have non-trivial residue on zero dimensional strata of the Baily-Borel compactification. A direct corollary is the non-vanishing of a higher regulator map.