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arXiv · 1507.07423

Elliptic curves with maximally disjoint division fields

Harris B. Daniels·Jeffrey Hatley·James Ricci
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Abstract

One of the many interesting algebraic objects associated to a given rational elliptic curve, $E$, is its full-torsion representation $ρ_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\mathrm{GL}_2(\hat{\mathbf{Z}})$. Generalizing this idea, one can create another full-torsion Galois representation, $ρ_{(E_1,E_2)}:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\left(\mathrm{GL}_2(\hat{\mathbf{Z}})\right)^2$ associated to a pair $(E_1,E_2)$ of rational elliptic curves. 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 $ρ_{(E_1,E_2)}$ 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 $ρ_{(E_1,E_2)}$ has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.

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Harris B. Daniels, Jeffrey Hatley, James Ricci. 2015-07-27. Elliptic curves with maximally disjoint division fields. https://arxiv.org/abs/1507.07423

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