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

TL;DR

This paper proves a surjectivity result for a map related to higher Chow groups and Chow groups of special fibers of products of elliptic curves over local fields, serving as a non-Archimedean analogue of the Hodge-D-conjecture.

Contribution

It establishes a non-Archimedean analogue of the Hodge-D-conjecture for products of elliptic curves, extending previous work to the case of non-isogenous semistable elliptic curves.

Findings

The map  is surjective for non-isogenous semistable elliptic curves.

Connects non-Archimedean Hodge- conjecture with higher Chow groups.

Extends known results to new cases of elliptic curves with split multiplicative reduction.

Abstract

In this paper we show that the map % $$\partial:CH^2(E_1 \times E_2,1)\otimes \Q \longrightarrow PCH^1(\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\XX_{v}$ of a semi-stable model $\XX$ of $E_1 \times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\D$-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.