On the equivalence of binary cubic forms

TL;DR

This paper establishes necessary and sufficient criteria for the equivalence of binary cubic forms over arbitrary fields, introducing the Cardano invariant and explicit transformation matrices, with applications to automorphisms and elliptic curve arithmetic.

Contribution

It provides new algebraic criteria for binary cubic form equivalence, including the Cardano invariant and explicit matrix constructions, extending previous work to arbitrary fields.

Findings

Derived criteria for form equivalence under GL(2,K) and SL(2,K)

Introduced the Cardano invariant related to classical formulas

Connected binary cubic forms to elliptic curve arithmetic

Abstract

We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either GL$(2,K)$ or SL$(2,K)$, deriving two necessary and sufficient criteria for such equivalence in each case. One of these involves an algebraic invariant of binary cubic forms which we call the Cardano invariant, which is closely connected to classical formulas and also appears in the work of Bhargava et al. The second criterion is expressed in terms of the base field itself, and also gives explicit matrices in SL$(2,K)$ or GL$(2,K)$ transforming one cubic into the other, if any exist, in terms of the coefficients of bilinear factors of a bicovariant of the two cubics. We also consider automorphisms of a single binary cubic form, show how to use our results to test equivalence of binary cubic forms over an integral domain such as~$\mathbb{Z}$, and briefly recall some connections between binary cubic forms and the arithmetic of elliptic curves. The methods used are elementary, and similar to those used in our earlier work with Fisher concerning equivalences between binary quartic forms.