#### Vol. 11, No. 5, 2017

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Modular curves of prime-power level with infinitely many rational points

### Andrew V. Sutherland and David Zywina

Vol. 11 (2017), No. 5, 1199–1229
##### Abstract

For each open subgroup $G$ of ${GL}_{2}\left(\stackrel{̂}{ℤ}\right)$ containing $-I$ with full determinant, let ${X}_{G}∕ℚ$ 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 ${X}_{G}\left(ℚ\right)$ 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 $\ell$-adic Galois representations arising from elliptic curves over $ℚ$ that is complete except for a finite set of exceptional $j$-invariants.

##### Keywords
modular curves, elliptic curves, Galois representations
##### Mathematical Subject Classification 2010
Primary: 14G35
Secondary: 11F80, 11G05

Group Tables