Algebra & Number Theory
- Algebra Number Theory
- Volume 11, Number 5 (2017), 1199-1229.
Modular curves of prime-power level with infinitely many rational points
For each open subgroup of containing with full determinant, let 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 . Up to conjugacy, we determine a complete list of the such groups of prime power level for which is infinite. For each , we also construct explicit maps from each to the -line. This list consists of modular curves of genus and modular curves of genus . 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 -invariants.
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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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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