Quadratic Chabauty for Modular Curves

TL;DR

This paper demonstrates that quadratic Chabauty can effectively determine rational points on certain modular curves of genus at least 3, especially when the Néron-Severi rank exceeds 1, extending classical methods.

Contribution

The paper proves that quadratic Chabauty is more effective than classical Chabauty for all modular curves of genus at least 3 with Néron-Severi rank at least 2.

Findings

Quadratic Chabauty is finite for these curves.

Classical Chabauty is less effective when rank conditions are not met.

Modular curves of genus ≥ 3 with high Néron-Severi rank benefit from quadratic Chabauty.

Abstract

Let $X/\mathbb{Q}$ be a curve of genus $g \ge 2$ with Jacobian $J$ and let $\ell$ be a prime of good reduction. Using Selmer varieties, Kim defines a decreasing sequence \[ X(\mathbb{Q}_\ell) \supseteq X(\mathbb{Q}_\ell)_1 \supseteq X(\mathbb{Q}_\ell)_2 \supseteq \cdots \] all containing the rational points of $X$. Thanks to the work of Coleman, the `Chabauty set' $X(\mathbb{Q}_\ell)_1$ is known to be finite provided the Mordell--Weil rank of $J$ is smaller than $g$. In this case one has a practical strategy that often succeeds in computing the set of rational points of $X$. Balakrishnan and Dogra have recently shown that the `quadratic Chabauty set' $X(\mathbb{Q}_\ell)_2$ is finite provided the Mordell--Weil rank is less than $g + \rho-1$, where $\rho$ is the N\'eron-Severi rank of $J/\mathbb{Q}$. In view of this it is interesting to give families of curves where $\rho \ge 2$ and where therefore quadratic Chabauty is more likely to succeed than classical Chabauty. In this note we show that this is indeed the case for all modular curves of genus at least 3.