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2017 Modular curves of prime-power level with infinitely many rational points
Andrew Sutherland, David Zywina
Algebra Number Theory 11(5): 1199-1229 (2017). DOI: 10.2140/ant.2017.11.1199


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.


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Andrew Sutherland. David Zywina. "Modular curves of prime-power level with infinitely many rational points." Algebra Number Theory 11 (5) 1199 - 1229, 2017.


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

zbMATH: 1374.14022
MathSciNet: MR3671434
Digital Object Identifier: 10.2140/ant.2017.11.1199

Primary: 14G35
Secondary: 11F80 , 11G05

Keywords: Elliptic curves , Galois representations , modular curves

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.11 • No. 5 • 2017
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