Modular curves of prime-power level with infinitely many rational points
By Andrew V. Sutherland, David Zywina. 2016
52 citations recorded by the source corpus.
For each open subgroup $G$ of ${\rm GL}_2(\hat{\mathbb{Z}})$ containing $-I$ with full determinant, let $X_G/\mathbb{Q}$ 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 $X_G(\mathbb{Q})$ 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 the $\ell$-adic Galois representations arising from elliptic curves over $\mathbb{Q}$ that is complete except for a finite set of exceptional $j$-invariants.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/121460f9b3db135f7fab25b73536274989284a49
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Galois representations attached to elliptic curves with complex multiplication (2018, background): “In [25] the authors provide a classification of all the possible p-adic representations over Q that is complete except for a finite set J of exceptional j-invariants (note that J includes all j-invariants in Q with CM).”