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Coclass of the second 3-class group

Siham Aouissi, Daniel C. Mayer

TL;DR

The paper establishes a coclass rule for the second $3$-class group $M$ of number fields with $ ext{Cl}_3(K)\simeq(3,3)$, showing that $cc(M)$ is determined by the second largest $3$-class number among the four unramified cyclic cubic extensions. It develops and deploys the Artin pattern and transfer-target/transfer-kernel framework (IPAD/TTT) to give explicit, nearly homocyclic invariant structures that predict $M$’s position in the coclass tree, including regular and irregular cases. The authors realize and catalog a wide range of arithmetic examples across imaginary and real quadratic fields, cyclic cubic fields, and pure cubic field closures, computing minimal discriminants and conductors and describing the corresponding normal lattices of the groups. These results advance explicit connections between metabelian $3$-groups and Hilbert $3$-class field towers, enabling systematic arithmetic realizations and providing a computational atlas of coclass phenomena in low-order $3$-groups.

Abstract

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application.

Coclass of the second 3-class group

TL;DR

The paper establishes a coclass rule for the second -class group of number fields with , showing that is determined by the second largest -class number among the four unramified cyclic cubic extensions. It develops and deploys the Artin pattern and transfer-target/transfer-kernel framework (IPAD/TTT) to give explicit, nearly homocyclic invariant structures that predict ’s position in the coclass tree, including regular and irregular cases. The authors realize and catalog a wide range of arithmetic examples across imaginary and real quadratic fields, cyclic cubic fields, and pure cubic field closures, computing minimal discriminants and conductors and describing the corresponding normal lattices of the groups. These results advance explicit connections between metabelian -groups and Hilbert -class field towers, enabling systematic arithmetic realizations and providing a computational atlas of coclass phenomena in low-order -groups.

Abstract

By means of parametrized presentations of finite metabelian 3-groups, it is proved that the coclass cc(M) of the second 3-class group M=Gal(F_3^2(K)/K) of any algebraic number field K with elementary bicyclic 3-class group Cl_3(K)=(3,3) is determined unambiguously by the second largest order ord(Cl_3(E_2))=3^{cc(M)+1} among the four 3-class groups of the unramified cyclic cubic extensions E_i (i=1,..,4) of K. Minimal discriminants of quadratic and cubic fields K with assigned coclass cc(M) are computed from extensive databases of 3-class numbers ord(Cl_3(E_i)) as an application.

Paper Structure

This paper contains 23 sections, 10 theorems, 25 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The coclass of a non-abelian 3-group
  3. The bi-polarization of a non-abelian 3-group
  4. Abelian type invariants and abelian quotient invariants
  5. The Artin pattern
  6. The main theorem
  7. Heterocyclic and homocyclic commutator groups
  8. Arithmetical realization of groups with assigned coclass
  9. Imaginary quadratic fields
  10. Real quadratic fields
  11. Cyclic cubic fields
  12. Normal closures of pure cubic fields
  13. Normal lattices of metabelian 3-groups
  14. Cyclic factor groups
  15. Bicyclic factor groups
  16. ...and 8 more sections

Key Result

Theorem 1

The coclass $\mathrm{cc}(M)$ of the second $3$-class group $M=\mathrm{Gal}(\mathrm{F}_3^2(K)/K)$ of an algebraic number field $K$ with elementary bicyclic $3$-class group $\mathrm{Cl}_3(K)\simeq(\mathbb{Z}/3\mathbb{Z})\times(\mathbb{Z}/3\mathbb{Z})$ and four unramified cyclic cubic extensions $E_i$$ where the $3$-class numbers $h_i=\#(\mathrm{Cl}_3(E_i))$ are arranged decreasingly $h_1\ge h_2\ge h

Figures (6)

  • Figure 1: Full normal lattice, including lower and upper central series, of a $p$-group $G$ with $G/G^\prime\simeq (p,p)$, $\mathrm{cl}(G)=c$, $\mathrm{cc}(G)=r$, $\mathrm{sl}(G)=2$, $k(G)=0$.
  • Figure 2: CF-groups $M=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ with cyclic factors only.
  • Figure 3: BF-groups $M=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ with bicyclic factors only.
  • Figure 4: Small BCF-groups $M=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ with bicyclic and cyclic factors.
  • Figure 5: Large BCF-groups $M=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ with bicyclic and cyclic factors.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Proposition 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 2
  • Theorem 2
  • ...and 16 more