Elliptic curves with maximally disjoint division fields

Harris B. Daniels¹ ; Jeffrey Hatley² ; James Ricci³

¹ Department of Mathematics Amherst College Box 2239 Amherst, MA 01002-5000, U.S.A.
² Department of Mathematics Union College Bailey Hall 202 Schenectady, NY 12308, U.S.A.
³ Department of Mathematics and Computer Science Daemen College Duns Scotus 339 4380 Main Street Amherst, NY 14226, U.S.A.

Acta Arithmetica, Tome 175 (2016) no. 3, pp. 211-223.

Résumé

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.

Détail

DOI : 10.4064/aa8275-7-2016

Keywords: many interesting algebraic objects associated given elliptic curve defined rational numbers mathbb its full torsion representation rho operatorname gal bar mathbb mathbb operatorname hat mathbb generalizing idea create another full torsion galois representation rho operatorname gal bar mathbb mathbb operatorname hat mathbb associated pair elliptic curves defined mathbb paper provide infinite number concrete examples pairs elliptic curves whose associated full torsion galois representation rho has maximal image size image inversely related size intersection various division fields defined representation rho has maximal image these division fields maximally disjoint paper devoted studying these intersections