A non-Archimedean analogue of the Hodge-$\mathcal {D}$-conjecture for products of elliptic curves

In this paper we show that the map \[ \partial :CH^2(E_1 \times E_2,1)\otimes \mathbb {Q} \longrightarrow PCH^1(\mathcal {X}_v)\] is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch’s higher Chow groups and $PCH^1(\mathcal {X}_v)$ is a certain subquotient of a Chow group of the special fibre $\mathcal {X}_{v}$ of a semi-stable model $\mathcal {X}$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\mathcal {D}$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis (J. Algebraic Geom. 14 (2005), 213–240), and on the other, an analogue of the works of Speiß ($K$-Theory 17 (1999), 363–383), Mildenhall (Duke Math. J. 67 (1992), 387–406) and Flach (Invent. Math. 109 (1992), 307–327) in the case when the elliptic curves have split multiplicative reduction.