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

TL;DR

This work advances Mazur's Program B for $\mathbb{Q}$ by developing an explicit, computable method to produce maps $X_G \to E$ from modular curves $X_G$ to rank $0$ elliptic curves, enabling effective determination of $X_G(\mathbb{Q})$ for a vast class of curves. Central to the approach is constructing a weight $2$ cusp form whose Hecke data matches a candidate $E$, computing period lattices to align $X_G$ with $E$, and realizing $x,y$ on $X_G$ as functions in the canonical ring to define a map to $E$, which then yields the rational points via pullback from $E(\mathbb{Q})$. The paper reports rigorous determination of $X_G(\mathbb{Q})$ for 797{,}591 modular curves of level $\le 70$, leaving only ten exceptional cases, and documents extensive computational data, including degrees of maps and CM/non-CM $j$-invariants. The results significantly enhance the understanding of rational points on modular curves and provide a practical pipeline for similar genus and level ranges, with potential extensions to higher ranks via Demjanenko–Manin methods or base-change analyses.

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 $70$.

Theorems & Definitions (6)

• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Proposition 5
• Proposition 6