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.