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Elliptic curves with maximally disjoint division fields

Volume 175 / 2016

Harris B. Daniels, Jeffrey Hatley, James Ricci Acta Arithmetica 175 (2016), 211-223 MSC: Primary 14H52; Secondary 11F80. DOI: 10.4064/aa8275-7-2016 Published online: 23 September 2016

Abstract

One of the many interesting algebraic objects associated to a given elliptic curve defined over the rational numbers, $E / \mathbb Q$, is its full-torsion representation $\rho_E:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\to\operatorname{GL}_2(\hat{\mathbb Z})$. Generalizing this idea, one can create another full-torsion Galois representation $\rho_{(E_1,E_2)}:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\to(\operatorname{GL}_2(\hat{\mathbb Z}))^2$ associated to a pair $(E_1,E_2)$ of elliptic curves defined over $\mathbb Q$. The goal of this paper is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation $\rho_{(E_1,E_2)}$ has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by $E_1$ and $E_2$. The representation $\rho_{(E_1,E_2)}$ has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.

Authors

  • Harris B. DanielsDepartment of Mathematics
    Amherst College
    Box 2239
    Amherst, MA 01002-5000, U.S.A.
    e-mail
  • Jeffrey HatleyDepartment of Mathematics
    Union College
    Bailey Hall 202
    Schenectady, NY 12308, U.S.A.
    e-mail
  • James RicciDepartment of Mathematics and Computer Science
    Daemen College
    Duns Scotus 339
    4380 Main Street
    Amherst, NY 14226, U.S.A.
    e-mail

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