Rational points on modular curves via maps to elliptic curves with rank zero

TL;DR

This paper develops a method to construct maps from modular curves to rank zero elliptic curves, enabling effective determination of rational points for most modular curves up to level 70, advancing understanding in arithmetic geometry.

Contribution

The authors introduce a novel approach to construct explicit maps from modular curves to elliptic curves of rank zero, facilitating the computation of rational points on these curves.

Findings

Successfully determined rational points on over 99% of modular curves up to level 70.

Provided a practical method for constructing maps to elliptic curves of rank zero.

Enhanced the toolkit for studying Galois representations associated with elliptic curves.

Abstract

A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$, there is a modular curve $X_G$ whose rational points parametrize elliptic curves for which the image of the mod $N$ Galois representation is contained in $G$. If $X_G$ admits a map to an elliptic curve $E / \mathbb{Q}$ of rank $0$, then its rational points can be effectively determined, provided such a map $X_G \to E$ is known. In this article, we give a method for constructing a map $X_G \to E$ and use it to determine $X_G(\mathbb{Q})$ for more than $99\%$ of modular curves of level at most $70$.