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

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 $E/\mathbb{Q}$ for which $E(\mathbb{Q})$ has rank $0$, then its rational points can be effectively determined, provided that a map $X_G \to E$ is known. In this article, we give a method for constructing such maps. Using this method, together with existing methods and results, we systematically determine the rational points of $X_G$ for more than $99\%$ of modular curves of level at most $70$.