Zero cycles on products of elliptic curves over local fields with supersingular reduction

TL;DR

This paper investigates zero cycles on products of supersingular elliptic curves over p-adic fields, producing infinitely many rational equivalences and providing evidence for a conjecture on the Albanese kernel's structure.

Contribution

It introduces a method to generate infinite rational equivalences on zero cycles using genus 2 covers, advancing understanding of the Albanese kernel in this context.

Findings

Infinite rational equivalences in Chow groups of zero cycles

Evidence supporting Colliot-Thélène's conjecture

Construction of zero cycles via genus 2 covers

Abstract

For a product $E_1\times E_2$ of two elliptic curves over a $p$-adic field with good supersingular reduction, we produce infinitely many rational equivalences in the Chow group $\mathrm{CH}_0(X)$ of zero cycles via genus 2 covers of $E_1$ and $E_2$. We use this to obtain evidence for a conjecture of Colliot-Th\'el\`ene about the structure of the Albanese kernel.