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

TL;DR

This paper develops methods to construct families of elliptic curves with specific mod 3 Galois representations, using classical geometric concepts like Hessian and Cayleyan curves, to understand their symplectic properties.

Contribution

It introduces a novel geometric approach to generate elliptic curves with prescribed mod 3 representations, distinguishing symplectic and anti-symplectic cases.

Findings

Constructed two families of elliptic curves with isomorphic mod 3 representations.

Established symplectic and anti-symplectic isomorphisms using Hessian and Cayleyan curves.

Provided explicit geometric methods for controlling mod 3 Galois representations.

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.