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

Sutherland, Andrew; Zywina, David

doi:10.2140/ant.2017.11.1199

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

Andrew V. Sutherland and David Zywina

Vol. 11 (2017), No. 5, 1199–1229

DOI: 10.2140/ant.2017.11.1199

Abstract

For each open subgroup $G$ of ${GL}_{2} (\hat{ℤ})$ 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 $X_{G} (ℚ)$ 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 $ℓ$ , 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.

Keywords

modular curves, elliptic curves, Galois representations

Mathematical Subject Classification 2010

Primary: 14G35

Secondary: 11F80, 11G05

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Received: 20 May 2016

Revised: 10 February 2017

Accepted: 10 March 2017

Published: 12 July 2017

Authors

Andrew V. Sutherland
Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, MA 02139 United States

David Zywina
Department of Mathematics Cornell University Ithaca, NY 14853 United States

Vol. 11, No. 5, 2017