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Smooth cuboids in group theory

Joshua Maglione, Mima Stanojkovski

TL;DR

The paper develops a geometric-algebraic framework for smooth cuboids, linking a $3 imes3$ matrix of linear forms with determinant a smooth cubic to a unipotent group scheme via the Baer correspondence. Central to the approach are adjoint algebras, which yield isomorphism invariants and a precise link between group isomorphism types and elliptic-curve data, including isomorphisms and isogenies. The authors derive explicit criteria for when two elliptic-group constructions are isomorphic, provide formulas for automorphism groups, and design concrete, Las Vegas–style algorithms for isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising from these structures, accompanied by a Magma implementation. This has its most immediate impact in computational group theory and arithmetic-geometry-inspired group classification, enabling efficient recognition and manipulation of otherwise intractable $p$-groups generated from elliptic curves. The work also clarifies how the arithmetic of the underlying elliptic curve, via $E[3]$ and flex points, governs the automorphism and isomorphism landscape of the associated $E$-groups.

Abstract

A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over $K$. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising in this way.

Smooth cuboids in group theory

TL;DR

The paper develops a geometric-algebraic framework for smooth cuboids, linking a matrix of linear forms with determinant a smooth cubic to a unipotent group scheme via the Baer correspondence. Central to the approach are adjoint algebras, which yield isomorphism invariants and a precise link between group isomorphism types and elliptic-curve data, including isomorphisms and isogenies. The authors derive explicit criteria for when two elliptic-group constructions are isomorphic, provide formulas for automorphism groups, and design concrete, Las Vegas–style algorithms for isomorphism testing of finite -groups of class and exponent arising from these structures, accompanied by a Magma implementation. This has its most immediate impact in computational group theory and arithmetic-geometry-inspired group classification, enabling efficient recognition and manipulation of otherwise intractable -groups generated from elliptic curves. The work also clarifies how the arithmetic of the underlying elliptic curve, via and flex points, governs the automorphism and isomorphism landscape of the associated -groups.

Abstract

A smooth cuboid can be identified with a matrix of linear forms, with coefficients in a field , whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over . We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite -groups of class and exponent arising in this way.
Paper Structure (28 sections, 40 theorems, 115 equations, 2 figures, 1 table)

This paper contains 28 sections, 40 theorems, 115 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Baer correspondence and Pfaffians
  4. Elliptic groups
  5. Elliptic groups and variations over the primes
  6. Groups from points on curves
  7. The contributions of this paper
  8. Tensors and unipotent group schemes
  9. Maps between $3$-tensors
  10. Algebras associated to $3$-tensors
  11. Structure of Artinian $*$-rings
  12. Unipotent group schemes and nilpotent Lie algebras from $3$-tensors
  13. Tensors and irreducible Pfaffians
  14. Pfaffians and isotropic subspaces
  15. Adjoint algebras of half-generic tensors
  16. ...and 13 more sections

Key Result

Theorem A

Let $F$ be a finite field with $\mathrm{char}(F)=p\geqslant 5$. Let, moreover, $E$ and $E'$ be elliptic curves in $\mathbb{P}^2_{F}$ given by Weierstrass equations, and let $P\in E(F)\setminus\{\mathcal{O}\}$ and $P'\in E'(F)\setminus\{\mathcal{O}'\}$. Then the following are equivalent.

Figures (2)

  • Figure 1: The runtimes of a Magma implementation of \ref{['mainthm:iso']} on the prime powers $p^e$, for $p\notin \{2,3\}$, up to $10^5$.
  • Figure 2: The isomorphism type for the $*$-semisimple part of the adjoint algebra for randomly constructed tensors.

Theorems & Definitions (97)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Corollary C
  • Remark 1.2
  • Theorem D
  • Theorem E
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • ...and 87 more