An Analogue of Gauss Composition for Binary Cubic Forms

TL;DR

This paper extends Gauss's composition law from binary quadratic forms to binary cubic forms using Bhargava's bijection, revealing a new explicit cubic analogue of Gaussian composition.

Contribution

It introduces an explicit form for a cubic analogue of Gaussian composition, connecting binary cubic forms with the 3-torsion of ideal class groups.

Findings

Established a cubic composition law analogous to Gauss's quadratic case.

Connected binary cubic forms with the 3-torsion subgroup of ideal class groups.

Provided explicit formulas for the composition law in the cubic setting.

Abstract

Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb Z})$-equivalence classes of primitive binary quadratic forms. Since then, bijections of classes of binary forms have been found with ideal class groups of quadratic rings. This paper uses one such bijection given by Bhargava, relating classes of projective binary cubic forms to the $3$-torsion of an ideal class group, to find an explicit form for a cubic analogue of Gaussian composition.