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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields
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by Jackson S. Morrow HTML | PDF
Math. Comp. 88 (2019), 2389-2421 Request permission

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
  • Email: jmorrow4692@gmail.com
  • Received by editor(s): September 4, 2017
  • Received by editor(s) in revised form: November 16, 2018
  • Published electronically: April 9, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 2389-2421
  • MSC (2010): Primary 11F80, 11G05; Secondary 11D45, 11G18
  • DOI: https://doi.org/10.1090/mcom/3426
  • MathSciNet review: 3957898