Vol. 11, No. 5, 2017

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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
Abstract

For each open subgroup $G$ of ${GL}_{2}\left(\stackrel{̂}{ℤ}\right)$ 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 ${X}_{G}\left(ℚ\right)$ 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 $\ell$, these results provide an explicit classification of the possible images of $\ell$-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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