Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves

2011/12/29 by Masato Kuwata, Kuwata, Masato
Mathematics · #11G05 #14H50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:14H50

paper · pdf · doi:10.48550/arxiv.1112.6317

After submitting the first version of this paper on the arXiv, the author was informed that the main observation of this paper had already been made by Tom Fisher, "The Hessian of a genus one curve", Proc. Lond. Math. Soc. (3) 104 (2012), 613-648 (arXiv:math/0610403v2)

arxiv created 2012/09/18 · arxiv updated 2012/09/19

Abstract

For a given elliptic curve E₀ defined over a number field k, we construct two families of elliptic curves whose mod 3 representations are isomorphic to that of E₀. The isomorphisms in the first family are symplectic, and those in the second family are anti-symplectic. Our construction is based on the notion of Hessian and Cayleyan curves in classical geometry.

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