On Indecomposable Elements of K1 of a Product of Elliptic Curves

Let E₁, E₂ be elliptic curves with good reduction over a local field k of residue characteristic p. Let X be the smooth projective model of E₁ × E₂ over the ring of integers of k. We show that KerCH²(X) CH² (E₁ × E₂)) is a finite p-group, by giving a new construction of indecomposable elements of H¹ _Zar(E₁ × E₂, K₂). As an application we show that the prime to p part of the torsion subgroup of CH²(E₁ × E₂) is finite.