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

TL;DR

This paper investigates the structure of zero-cycles on products of elliptic curves over p-adic fields, extending existing results to include cases with supersingular reduction and providing criteria for cycle map injectivity.

Contribution

It extends the understanding of the Albanese kernel for products of elliptic curves to cases with supersingular reduction and establishes criteria for cycle map injectivity over finite extensions.

Findings

Albanese kernel is a sum of finite and divisible groups under certain reduction conditions.

Criteria for injectivity of the cycle map are provided for all n ≥ 1.

Extension of previous results to include supersingular phenomena.

Abstract

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.