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

TL;DR

The paper addresses Colliot-Thélène's conjecture on the Albanese kernel $F^2$ for a product of supersingular elliptic curves over a p-adic field by analyzing the first level of the Somekawa $K$-group modulo $p$. It develops a framework based on WR relations and the signature of $K/K$-symbols, and constructs infinitely many WR relations from Scholten genus-2 curves that span the pair of elliptic curves, achieving diagonal signatures $(n,n)$ for all $n$ in the totally ramified setting with $p>3$. This diagonal-signature strategy provides a mechanism to force vanishing of the $p$-torsion part of the Somekawa group, offering evidence toward the Colliot-Thélène conjecture in the supersingular–supersingular case and recovering earlier ordinary/supersingular results as special cases. The paper also includes computational evidence for quadratic ramified extensions, using a Sage-based approach to generate explicit genus-2 spans and verify symbol vanishing in many instances, suggesting robustness of the method beyond theoretical guarantees in all cases.

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-Thélène about the structure of the Albanese kernel.

Theorems & Definitions (41)

• Conjecture A
• Theorem A
• Theorem 1.1
• Remark 1.1
• Definition 1.2
• Proposition 1.3
• proof
• Definition 1.4
• Remark 1.2
• Lemma 1.1