Zero-cycles on a product of elliptic curves over a $p$-adic field

We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ 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 of Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $CH_0(X)/p^n\rightarrow H^{2d}_{\text{\'{e}t}}(X, \mu_{p^n}^{\otimes d})$. We give specific criteria that guarantee this map is injective for every $n\geq 1$. 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 of Yamazaki and Hiranouchi.