Algebra & Number Theory

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

Andrew Sutherland and David Zywina

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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.

Article information

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

Received: 20 May 2016
Revised: 10 February 2017
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Secondary: 11F80: Galois representations 11G05: Elliptic curves over global fields [See also 14H52]

modular curves elliptic curves Galois representations


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.

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Supplemental materials

  • Group Tables.