Zero Cycles on a Product of Elliptic Curves Over a p-adic Field

We consider a product X = E-1 x ... x E-d of elliptic curves over a finite extension K of Q(p) with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of X is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map CH0(X)/p(n) -> H-et(2d)(X, mu(circle times d)(pn)). We give specific criteria that guarantee this map is injective for every n >= 1. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension L of K for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi.