Regulator indecomposable cycles on a product of elliptic curves (Q2861285)

The paper under review, which was authored by a student under the supervision of the reviewer, deals with the issue of detecting indecomposable \(K_1\) classes on a general product \(E_1\times E_2\) of elliptic curves. An earlier paper, authored by the reviewer and \textit{B. B. Gordon} [J. Algebr. Geom. 8, No. 3, 543--567 (1999; Zbl 0940.14001)], attempted to solve this problem by a degeneration argument involving a singular rational surface. That technique contained an error -- and a subsequent paper by the author and \textit{Xi Chen}, in the same journal [J. Algebr. Geom. 14, No. 2, 213--240 (2005; Zbl 1075.14005)], settled the problem, albeit using entirely different techniques. The reviewer decided that the original paper could be fixed, under the same idea of techniques, by working with the same pre-\(K_1\)-class involving a pair \((f,D)\), where \(D\subset E_1\times E_2\) is a particular surface, and \(f\in {\mathbb C}(E_1\times E_2)^{\times}\) is modified so that the original degeneration leads to a non-trivial current defined by \(\int_D\log|f|\wedge(-)\). Extending \((f,D)\) to an indecomposable \(K_1\) class was the problem. However, by a dimension count, one can approximate \(f\) by a quotient of two quadric polynomials (call it \(\tilde{f}\) on \(D\)), for which the ``precycle'' \((\tilde{f},D)\) can be completed to an indecomposable class in \(K_1\). The technique makes heavy use of the existence of torsion points on elliptic curves.