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Explicit composition identities for higher composition laws

Gautam Chinta, Ajith Nair

TL;DR

This work provides explicit Gauss-like composition identities for Bhargava’s five higher composition laws, linking each space (2×2×2 cubes, binary cubic forms, pairs of binary quadratic forms, pairs of quaternary alternating 2-forms, and senary alternating 3-forms) to discriminant-preserving ideal-class data in oriented quadratic rings. The approach generalizes Gauss’s classic form composition by giving multilinear, bilinear, or covariant reformulations via cube- and form-products, companion objects, and dual constructions, with precise compatibility between the quadratic forms and the ideal data. The results yield concrete, computable formulas for the group structure on projective orbits, and reveal interpretations in terms of narrow class groups, balanced ideals, and associated covariants, providing new tools for arithmetic investigations and potential field-counting applications. Overall, the paper unifies and extends Bhargava’s framework by delivering explicit, Gauss-type identities across multiple higher-dimensional spaces, maintaining discriminant invariance throughout.

Abstract

In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces: 1) binary cubic forms with triplicate middle coefficients, 2) pairs of binary quadratic forms with duplicate middle coefficients, 3) pairs of quaternary alternating 2-forms and 4) senary alternating 3-forms. In each of the five cases, there is a natural group action on the underlying space with a unique polynomial invariant called the discriminant, and a notion of projectivity for the elements of the space. The strategy behind Bhargava's approach is to construct a discriminant-preserving bijection between the set of orbits under the group action and the set of (tuples of) suitable ideal classes of quadratic rings. The projective ideal classes are equipped with a natural group structure and hence we get a group structure on the spaces of equivalence classes of projective forms of fixed discriminant $D$. In each case the class group of projective forms of discriminant $D$ has a natural interpretation in terms of the narrow class group of the quadratic ring of discriminant $D$. The aim of this paper is to give explicit composition identities (similar to Gauss' formulation of composition of binary quadratic forms) for these higher composition laws.

Explicit composition identities for higher composition laws

TL;DR

This work provides explicit Gauss-like composition identities for Bhargava’s five higher composition laws, linking each space (2×2×2 cubes, binary cubic forms, pairs of binary quadratic forms, pairs of quaternary alternating 2-forms, and senary alternating 3-forms) to discriminant-preserving ideal-class data in oriented quadratic rings. The approach generalizes Gauss’s classic form composition by giving multilinear, bilinear, or covariant reformulations via cube- and form-products, companion objects, and dual constructions, with precise compatibility between the quadratic forms and the ideal data. The results yield concrete, computable formulas for the group structure on projective orbits, and reveal interpretations in terms of narrow class groups, balanced ideals, and associated covariants, providing new tools for arithmetic investigations and potential field-counting applications. Overall, the paper unifies and extends Bhargava’s framework by delivering explicit, Gauss-type identities across multiple higher-dimensional spaces, maintaining discriminant invariance throughout.

Abstract

In 2001, Bhargava proved a composition law for integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces: 1) binary cubic forms with triplicate middle coefficients, 2) pairs of binary quadratic forms with duplicate middle coefficients, 3) pairs of quaternary alternating 2-forms and 4) senary alternating 3-forms. In each of the five cases, there is a natural group action on the underlying space with a unique polynomial invariant called the discriminant, and a notion of projectivity for the elements of the space. The strategy behind Bhargava's approach is to construct a discriminant-preserving bijection between the set of orbits under the group action and the set of (tuples of) suitable ideal classes of quadratic rings. The projective ideal classes are equipped with a natural group structure and hence we get a group structure on the spaces of equivalence classes of projective forms of fixed discriminant . In each case the class group of projective forms of discriminant has a natural interpretation in terms of the narrow class group of the quadratic ring of discriminant . The aim of this paper is to give explicit composition identities (similar to Gauss' formulation of composition of binary quadratic forms) for these higher composition laws.
Paper Structure (29 sections, 16 theorems, 186 equations)

This paper contains 29 sections, 16 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Cube Law and dual cubes
  4. Explicit composition of cubes
  5. Summary of main results and outline of paper
  6. Gauss composition of binary quadratic forms
  7. Composition of 2 x 2 x 2 cubes
  8. Correspondence between 2 x 2 x 2 cubes and triples of ideals
  9. Composition identities for 2 x 2 x 2 cubes
  10. Proof of Theorem \ref{['thm:cubecomp']}, forward direction
  11. Proof of Theorem \ref{['thm:cubecomp']}, reverse direction.
  12. Example
  13. Composition of binary cubic forms
  14. The group law for binary cubic forms
  15. Explicit composition for binary cubic forms
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $A, B, C\in \mathbb{Z}^2\otimes\mathbb{Z}^2\otimes\mathbb{Z}^2$ be three projective cubes of common discriminant $D$ and $\epsilon\in\{0,1\}$ congruent to $D$ mod 4. Let $\underline{x},\underline{y},\underline{z},\underline{u},\underline{v},\underline{w}\in \mathbb{Z}^2.$ Then if and only if there exist cubes $R=(r_{ijk}),S=(s_{ijk}),T=(t_{ijk})$ of discriminant $D$ satisfying and and If t

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2: bhargava:2004, Eq. (20)
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 20 more