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Mathematics of Computation

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Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields

Author: Jackson S. Morrow
Journal: Math. Comp. 88 (2019), 2389-2421
MSC (2010): Primary 11F80, 11G05; Secondary 11D45, 11G18
Published electronically: April 9, 2019
MathSciNet review: 3957898
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Abstract: Let $E$ be an elliptic curve defined over $\mathbf {Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho _{E,\ell }\colon \operatorname {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \operatorname {GL}_2(\mathbf {Z}/\ell \mathbf {Z})$ that describes the Galois action on the $\ell$-torsion points of $E$. Building on recent work of Rouse–Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\mathbf {Q}$ with simultaneously non-surjective, composite images of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields.

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Additional Information

Jackson S. Morrow
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
MR Author ID: 1099817

Received by editor(s): September 4, 2017
Received by editor(s) in revised form: November 16, 2018
Published electronically: April 9, 2019
Article copyright: © Copyright 2019 American Mathematical Society