Abelian surfaces and the non-Archimedean Hodge D-conjecture -- the semi-stable case

TL;DR

This paper investigates a non-Archimedean analogue of the Hodge D-conjecture for Abelian surfaces with bad reduction, replacing Deligne cohomology with Chow groups of the special fibre, extending known results to more complex cases.

Contribution

It proves an analogue of the Hodge D-conjecture for Abelian surfaces at non-Archimedean places with bad reduction, expanding the understanding of the conjecture in non-Archimedean settings.

Findings

Established surjectivity of the regulator map in the bad reduction case

Extended previous results from elliptic curves to Abelian surfaces

Provided new insights into the non-Archimedean Hodge D-conjecture

Abstract

If $X$ is a smooth projective variety over ${\mathbb R}$, the Hodge ${\mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess in the case of products of elliptic curve and by me in general.