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On the equivalence of binary cubic forms

J E Cremona

TL;DR

This work provides two complete, explicit criteria for testing $ ext{GL}(2,K)$- and $ ext{SL}(2,K)$-equivalence of binary cubic forms over fields with $ ext{char} eq2,3$: a Cardano invariant $z(g)$ living in the resolvent algebra $L=K[\delta]$ with $\delta^2=-3\Delta$, and a bicovariant-based criterion that detects equivalence via bilinear factors of the cubic bicovariant $B_{g_1,g_2}$. The Cardano criterion partitions $ ext{SL}(2,K)$-orbits via the Cardano group $(L^*/L^{*3})_{N=1}$ and yields explicit SL(2,K) or GL(2,K) transformation data when equivalence exists. The bicovariant approach provides an alternative, purely algebraic path to find the transforming matrices, with precise control over determinant and the number of possible correspondences. The results tie into integral equivalence and elliptic-curve arithmetic, notably describing connections to $3$-isogeny descent and the Delone–Faddeev/ Bhargava–Elkies–Shnidman framework for cubic forms.

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.

On the equivalence of binary cubic forms

TL;DR

This work provides two complete, explicit criteria for testing - and -equivalence of binary cubic forms over fields with : a Cardano invariant living in the resolvent algebra with , and a bicovariant-based criterion that detects equivalence via bilinear factors of the cubic bicovariant . The Cardano criterion partitions -orbits via the Cardano group and yields explicit SL(2,K) or GL(2,K) transformation data when equivalence exists. The bicovariant approach provides an alternative, purely algebraic path to find the transforming matrices, with precise control over determinant and the number of possible correspondences. The results tie into integral equivalence and elliptic-curve arithmetic, notably describing connections to -isogeny descent and the Delone–Faddeev/ Bhargava–Elkies–Shnidman framework for cubic forms.

Abstract

We consider the question of determining whether two binary cubic forms over an arbitrary field whose characteristic is not or are equivalent under the actions of either GL or SL, 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 or GL 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~, 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.
Paper Structure (9 sections, 18 theorems, 33 equations)

This paper contains 9 sections, 18 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Binary cubics, their invariants and covariants
  3. The resolvent algebra and the Cardano invariant
  4. Equivalence of binary cubics
  5. Cubic equivalence in terms of equality of Cardano invariants
  6. Cubic equivalence in terms of factors of a cubic bicovariant
  7. Automorphisms of cubic forms
  8. Integral equivalence
  9. Applications to elliptic curves

Key Result

Theorem 1

Let $K$ be any field with $\mathop{\mathrm{char}}\nolimits(K)\not=2,3$. Let $\Delta\in K^*$, let $L$ be the resolvent algebra $K[X]/(X^2+3\Delta)$, and let $z\colon {\mathcal{BC}}(K;\Delta)\to L^*/L^*{}^3$ be the Cardano invariant map.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Definition 1
  • Proposition 3
  • proof
  • Corollary 4
  • proof
  • Proposition 5
  • ...and 29 more