## Algebra & Number Theory

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

#### Abstract

For each open subgroup $G$ of $GL2(ℤ ̂)$ containing $− I$ with full determinant, let $XG∕ℚ$ 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 $XG(ℚ)$ is infinite. For each $G$, we also construct explicit maps from each $XG$ to the $j$-line. This list consists of $220$ modular curves of genus $0$ and $28$ modular curves of genus $1$. For each prime $ℓ$, these results provide an explicit classification of the possible images of $ℓ$-adic Galois representations arising from elliptic curves over $ℚ$ that is complete except for a finite set of exceptional $j$-invariants.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 5 (2017), 1199-1229.

Dates
Revised: 10 February 2017
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513090725

Digital Object Identifier
doi:10.2140/ant.2017.11.1199

Mathematical Reviews number (MathSciNet)
MR3671434

Zentralblatt MATH identifier
1374.14022

#### Citation

Sutherland, Andrew; Zywina, David. Modular curves of prime-power level with infinitely many rational points. Algebra Number Theory 11 (2017), no. 5, 1199--1229. doi:10.2140/ant.2017.11.1199. https://projecteuclid.org/euclid.ant/1513090725

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#### Supplemental materials

• Group Tables.