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

Abstract

For a given elliptic curve $E_0$ defined over a number field $k$, we construct two families of elliptic curves whose mod 3 representations are isomorphic to that of $E_0$. 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.

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)

Cited by in corpus (1)

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