Abelian surfaces and the non-Archimedean Hodge-$\D$-conjecture – The semi-stable case

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.