Explicit Motivic Mixed Elliptic Chabauty-Kim

TL;DR

This paper extends the explicit motivic Chabauty-Kim method to non-rational curves, especially punctured elliptic curves, by developing an explicit Tannakian framework using p-adic Galois representations.

Contribution

It introduces an explicit Tannakian Chabauty-Kim method applicable to higher genus curves, broadening the scope of the original approach.

Findings

Calculated the form of the Chabauty-Kim ideal for punctured elliptic curves.

Developed a Tannakian framework using p-adic Galois representations.

Laid groundwork for applying the method to higher genus curves.

Abstract

The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of an element of the Chabauty-Kim ideal for $\mathbb{Z}[1/\ell]$-points on a punctured elliptic curve, and lay some groundwork for certain kinds of higher genus curves. For this purpose, we develop an "explicit Tannakian Chabauty-Kim method" using $\mathbb{Q}_{p}$-Tannakian categories of Galois representations in place of $\mathbb{Q}$-linear motives. In future work, we intend to use this method to explicitly apply the Chabauty-Kim method to a curve of positive genus in a situation where Quadratic Chabauty does not apply.