Elliptic curves with maximally disjoint division fields

2015/07/27 by Daniels, Harris B., Hatley, Jeffrey, Ricci, James
#11F80 #14H52 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1507.07423

Abstract

One of the many interesting algebraic objects associated to a given rational elliptic curve, E, is its full-torsion representation ρ_E:Gal(Q/Q)\toGL₂(Z). Generalizing this idea, one can create another full-torsion Galois representation, ρ_(E1,E2):Gal(Q/Q)→(GL₂(Z))² associated to a pair (E₁,E₂) 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_(E1,E2) has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by E₁ and E₂. The representation ρ_(E1,E2) 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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