Real Regulators for Products of Elliptic Curves

TL;DR

This paper demonstrates the failure of the Hodge-${\mathcal D}$-conjecture for certain real regulators on products of general elliptic curves, assuming a specific Chow group decomposition.

Contribution

It proves the conjecture fails for products of elliptic curves under the Kunneth decomposition assumption, revealing limitations in current regulator theories.

Findings

Hodge-${\mathcal D}$-conjecture fails for certain products of elliptic curves

Failure occurs when 2n ≥ 3k - 1 ≥ 8

Results depend on the Kunneth decomposition assumption

Abstract

Assuming the Kunneth decomposition of the Chow groups of products of general Kummer surfaces, we prove that the Hodge-${\mathcal D}$-conjecture fails for the real regulator $r_{k,1}$ on a product of $n$ general elliptic curves for $2n\ge 3k-1\ge 8$.