Modular curves of prime-power level with infinitely many rational points

Abstract

For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ denote the modular curve that loosely parametrizes elliptic curves whose Galois representation, which arises from the Galois action on its torsion points, has image contained in $G$. Up to conjugacy, we determine a complete list of the $248$ such groups $G$ of prime power level for which $X_G(\mathbb{Q})$ is infinite. For each $G$, we also construct explicit maps from each $X_G$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $\ell$, these results provide an explicit classification of the possible images of the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.