A non-Archimedean analogue of the Hodge-D-conjecture for products of elliptic curves

In this paper we show that the map % $$\mathrm{\partial }:C{H}^2(E_1\times E_2,1)\otimes \text{Q}⟶PC{H}^1({\text{XX}}_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$isoneofBloc{h}^{\mathrm{\prime }}shigherChowgroupsand$PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre ${\text{XX}}_v$ of a semi-stable model $\text{XX}$ of $E_1\times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\text{D}$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \cite{lech}, and on the other, an analogue of the works of Spei{\ss} \cite{spie}, Mildenhall \cite{mild} and Flach \cite{flac} in the case when the elliptic curves have split multiplicative reduction.