Vol. 11, No. 5, 2017

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 GL2( ̂) containing I with full determinant, let XG 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() 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 , these results provide an explicit classification of the possible images of -adic Galois representations arising from elliptic curves over 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