The strategy of pattern recognition via Artin transfers applied to finite towers of 2-class fields

TL;DR

This paper develops a pattern recognition strategy using Artin transfers to determine the Galois group structure of 2-class field towers in quadratic fields with specific class group invariants, advancing understanding of their tower lengths.

Contribution

It introduces a novel approach combining Artin pattern analysis with p-group generation to classify metabelian 2-groups in 2-class field towers, extending existing classifications.

Findings

Determined Galois group types for specific quadratic fields.

Linked tower length to 2-class group rank.

Enhanced classification methods using pattern recognition.

Abstract

The isomorphism type of the Galois group of the 2-class field tower of quadratic number fields having a 2-class group with abelian type invariants (4,4) is determined by means of information on the transfer of 2-classes to unramified abelian 2-extensions, collected in the Artin pattern. In recent investigations by Benjamin and Snyder, the length of the tower of such fields has turned out to be dependent on the rank of the 2-class group of the first Hilbert 2-class field. Significant progress is achieved by extending the pool of possible metabelian 2-groups of the second Hilbert 2-class field from the SmallGroups database, resp. Hall-Senior classification, with the aid of the p-group generation algorithm, and sifting the pool by means of pattern recognition.

Tables

Table 1. Table 1. Discriminants of imaginary quadratic fields K 𝐾 K with ℓ 2 ​ ( K ) = 2 subscript ℓ 2 𝐾 2 \ell_{2}(K)=2

$d_K$	Factorization	$\mathrm{Gal}\left(F_2^{\infty }\left(K\right)/K\right)$	${\tau }_4$
$-\mathrm{6\hspace{0.17em}052}$	$\left(-4\right)\cdot 17\cdot 89$	$⟨128,31⟩$	$\left(21\right)$
$-\mathrm{8\hspace{0.17em}103}$	$\left(-3\right)\cdot 37\cdot 73$	$⟨256,285⟩$	$\left(31\right)$
$-\mathrm{11\hspace{0.17em}972}$	$\left(-4\right)\cdot 41\cdot 73$	$⟨256,286⟩$	$\left(31\right)$
$-\mathrm{14\hspace{0.17em}547}$	$\left(-3\right)\cdot 13\cdot 373$	$⟨512,1605⟩-\#1;2$	$\left(51\right)$
$-\mathrm{17\hspace{0.17em}476}$	$\left(-4\right)\cdot 17\cdot 257$	$⟨512,1598⟩$	$\left(41\right)$

Table 2. Table 2. Comparison of the candidates for Artin pattern ( 4.2 )

Vertex	${\tau }_1$	${\varkappa }_1$	${\tau }_2$	${\varkappa }_2$	${\tau }_4$
$⟨128,26⟩$	$211,221,311$	$J_0,K_3,J_0$	$22,22,31,31,211,211;2111$	$H_3,H_3,G,G,G,G;H_3$	$\left(21\right)$
$⟨128,31⟩$	$211,221,311$	$J_0,K_2,K_2$	$22,22,31,31,211,211;221$	$H_2,H_2,H_2,H_2,H_3,H_3;H_2$	$\left(21\right)$
$⟨128,34⟩$	$211,221,311$	$J_0,K_3,K_2$	$22,22,31,31,31,31;221$	$H_3,H_3,H_2,H_2,H_2,H_2;J_0$	$\left(21\right)$
$⟨128,35⟩$	$211,221,311$	$J_0,K_1,J_0$	$22,22,31,31,31,31;221$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(21\right)$

Table 3. Table 3. Comparison of the candidates for Artin pattern ( 4.8 )

Vertex	${\varkappa }_1$	${\varkappa }_2$	${\tau }_4$
$⟨512,1595⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,G,G;H_2$	$\left(41\right)$
$⟨512,1596⟩$	$J_0,K_2,K_3$	$H_1,H_1,H_2,H_2,H_3,H_3;J_0$	$\left(41\right)$
$⟨512,1597⟩$	$J_0,K_2,K_1$	$H_1,H_1,H_2,H_2,H_1,H_1;J_0$	$\left(41\right)$
$⟨512,1598⟩$	$J_0,K_2,K_2$	$H_1,H_1,H_2,H_2,H_2,H_2;H_2$	$\left(41\right)$

Table 4. Table 4. States with increasing polarized transfer targets

State	Order	Identifiers	${\tau }_1\left(3\right)$	${\tau }_2\left(5\right)$	${\tau }_2\left(7\right)$	${\tau }_4$
Root	$128$	$35$	$\left(311\right)$	$\left(31\right)$	$\left(221\right)$	$\left(21\right)$
Ground	$256$	$284,285,286,287$	$\left(411\right)$	$\left(41\right)$	$\left(321\right)$	$\left(31\right)$
1st Excited	$512$	$1605,1606,1607,1608$	$\left(511\right)$	$\left(51\right)$	$\left(421\right)$	$\left(41\right)$
2nd Excited	$1024$	$1,2,3,4$	$\left(611\right)$	$\left(61\right)$	$\left(521\right)$	$\left(51\right)$

Table 5. Table 5. Variants with distinct polarized transfer kernels

Variant	Identifiers	${\varkappa }_1\left(3\right)$	${\varkappa }_2\left(5\right)$	${\varkappa }_2\left(7\right)$
Mainline	$⟨128,35⟩,⟨256,284⟩,⟨512,1605⟩,1$	$J_0$	$G$	$H_1$
First	$⟨256,285⟩,⟨512,1606⟩,2$	$K_3$	$H_3$	$J_0$
Second	$⟨256,286⟩,⟨512,1607⟩,3$	$K_1$	$H_1$	$H_1$
Third	$⟨256,287⟩,⟨512,1608⟩,4$	$K_2$	$H_2$	$J_0$

Table 6. Table 6. Discriminants of imaginary quadratic fields K 𝐾 K with ℓ 2 ​ ( K ) = 3 subscript ℓ 2 𝐾 3 \ell_{2}(K)=3

$d_K$	Factorization	$\mathrm{Gal}\left(F_2^2\left(K\right)/K\right)$	${\tau }_4$
$-\mathrm{6\hspace{0.17em}123}$	$\left(-3\right)\cdot 13\cdot 157$	$⟨512,227⟩-\#2;140$	$\left(421\right)$
$-\mathrm{25\hspace{0.17em}144}$	$8\cdot \left(-7\right)\cdot 449$	$⟨512,231⟩-\#1;4$	$\left(321\right)$

Table 7. Table 7. Development of the Artin pattern along the root path

Vertex	${\tau }_1$	${\varkappa }_1$	${\tau }_2$	${\varkappa }_2$	${\tau }_4$
$⟨16,2⟩$	$21,21,21$	$J_0,J_0,J_0$	$2,2,2,2,2,2;11$	$G,G,G,G,G,G;G$	$\left(0\right)$
$⟨32,2⟩$	$211,211,211$	$J_0,J_0,J_0$	$21,21,21,21,21,21;111$	$G,G,G,G,G,G;G$	$\left(1\right)$
$⟨128,26⟩$	$211,221,311$	$J_0,K_3,J_0$	$22,22,31,31,211,211;2111$	$H_3,H_3,G,G,G,G;H_3$	$\left(21\right)$
$⟨512,231⟩$	$211,221,411$	$J_0,K_3,K_1$	$22,22,41,41,221,2111;3111$	$H_3,H_3,H_1,H_1,G,H_1;J_0$	$\left(311\right)$
$\#1;3$	$211,221,411$	$J_0,K_3,K_1$	$22,22,41,41,222,2111;3111$	$H_3,H_3,H_1,H_1,H_1,H_1;J_0$	$\left(321\right)$
$\#1;4$	$211,221,411$	$J_0,K_3,K_1$	$22,22,41,41,222,2111;3111$	$H_3,H_3,H_1,H_1,H_2,H_1;J_0$	$\left(321\right)$
$\#1;7$	$211,221,411$	$J_0,K_3,K_1$	$22,22,41,41,222,2111;3111$	$H_3,H_3,H_1,H_1,G,H_1;J_0$	$\left(321\right)$
$\#1;8$	$211,221,411$	$J_0,K_3,K_1$	$22,22,41,41,222,2111;3111$	$H_3,H_3,H_1,H_1,H_3,H_1;J_0$	$\left(321\right)$

Table 8. Table 8. Selected higher abelian quotient invariants of known groups and K 𝐾 K

Group resp. Field	${\tau }_{2,1}$
$M=⟨512,227⟩-\#2;140$	$[{(22;211,211,411)}^2,{(51;51,51,411)}^2,(2111;211,\dots ),\dots ]$
$⟨512,227⟩-\#3;39$	$[{(22;221,221,411)}^2,{(51;52,52,411)}^2,(2111;2111,\dots ),\dots ]$
${𝔊}_{20}$	$[{(22;221,221,411)}^2,{(51;52,52,411)}^2,(2111;2211,\dots ),\dots ]$
$K=\mathbb{Q}\left(\sqrt{-6123}\right)$	$[{(22;311,311,411)}^2,{(51;62,62,411)}^2,(2111;3111,\dots ),\dots ]$

Table 9. Table 9. Discriminant of an imaginary quadratic field K 𝐾 K with ℓ 2 ​ ( K ) = 2 subscript ℓ 2 𝐾 2 \ell_{2}(K)=2

$d_K$	Factorization	$\mathrm{Gal}\left(F_2^2\left(K\right)/K\right)$	${\tau }_4$
$-\mathrm{2\hspace{0.17em}379}$	$\left(-3\right)\cdot 13\cdot 61$	$⟨256,28⟩-\#2;i-\#1;j$	$\left(421\right)$

Table 10. Table 10. Discriminants of real quadratic fields K 𝐾 K with ℓ 2 ​ ( K ) = 2 subscript ℓ 2 𝐾 2 \ell_{2}(K)=2

$d_K$	Factorization	$\mathrm{Gal}\left(F_2^{\infty }\left(K\right)/K\right)$	${\tau }_4$
$\mathrm{58\hspace{0.17em}888}$	$8\cdot 17\cdot 433$	$⟨32,4⟩$	$\left(1\right)$
$\mathrm{84\hspace{0.17em}972}$	$4\cdot 3\cdot 73\cdot 97$	$⟨64,20⟩$	$\left(2\right)$
$\mathrm{257\hspace{0.17em}045}$	$5\cdot 101\cdot 509$	$⟨128,19⟩$	$\left(21\right)$
$\mathrm{232\hspace{0.17em}328}$	$8\cdot 113\cdot 257$	$⟨256,281⟩$	$\left(31\right)$
$\mathrm{26\hspace{0.17em}245}$	$5\cdot 29\cdot 181$	$⟨256,210\text{ or }211⟩$	$\left(31\right)$

Table 11. Table 11. Comparison of the contestants for Artin pattern ( 5.11 )

Vertex	${\varkappa }_1$	${\varkappa }_2$	${\tau }_4$	$\zeta$
$⟨256,280⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,H_3,H_3;H_2$	$\left(31\right)$	$\left(11\right)$
$⟨256,281⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,G,G;J_0$	$\left(31\right)$	$\left(11\right)$
$⟨256,282⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,H_2,H_2;J_0$	$\left(31\right)$	$\left(11\right)$
$⟨256,283⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,H_1,H_1;H_2$	$\left(31\right)$	$\left(11\right)$
$⟨512,1603⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,H_3,H_3;H_2$	$\left(41\right)$	$\left(1\right)$
$⟨512,1604⟩$	$J_0,K_2,J_0$	$H_1,H_1,H_2,H_2,H_3,H_3;H_2$	$\left(41\right)$	$\left(1\right)$

Table 12. Table 12. Comparison of the contestants for Artin pattern ( 5.13 )

Vertex	${\varkappa }_1$	${\varkappa }_2$	${\tau }_4$	$\zeta$
$⟨256,209⟩$	$J_0,J_0,J_0$	$H_1,H_1,G,G,H_3,H_3;J_0$	$\left(31\right)$	$\left(11\right)$
$⟨256,210⟩$	$J_0,J_0,J_0$	$H_1,H_1,G,G,H_3,H_3;H_2$	$\left(31\right)$	$\left(11\right)$
$⟨256,211⟩$	$J_0,J_0,J_0$	$H_1,H_1,G,G,H_3,H_3;H_2$	$\left(31\right)$	$\left(11\right)$
$⟨256,212⟩$	$J_0,J_0,J_0$	$H_1,H_1,G,G,H_3,H_3;J_0$	$\left(31\right)$	$\left(11\right)$

Table 13. Table 13. Discriminant of a real quadratic field K 𝐾 K with ℓ 2 ​ ( K ) = 2 subscript ℓ 2 𝐾 2 \ell_{2}(K)=2

$d_K$	Factorization	$\mathrm{Gal}\left(F_2^2\left(K\right)/K\right)$	${\tau }_4$
$\mathrm{595\hspace{0.17em}561}$	$17\cdot 53\cdot 661$	$⟨512,1465⟩-\#1;2$	$\left(222\right)$

Table 14. Table 14. Development of the Artin pattern along the root path

Vertex	${\tau }_1$	${\varkappa }_1$	${\tau }_2$	${\varkappa }_2$	${\tau }_4$
$⟨16,2⟩$	$21,21,21$	$J_0,J_0,J_0$	$2,2,2,2,2,2;11$	$G,G,G,G,G,G;G$	$\left(0\right)$
$⟨64,19⟩$	$211,211,211$	$J_0,J_0,J_0$	$31,31,31,31,31,31;211$	$H_3,H_3,H_2,H_2,H_1,H_1;G$	$\left(2\right)$
$⟨128,25⟩$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,311;2111$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(21\right)$
$⟨256,237⟩$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,321;2111$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(211\right)$
$⟨512,1465⟩$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,322;2111$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(221\right)$
$\#1;1$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,422;2111$	$H_3,H_3,H_2,H_2,H_1,J_{12};H_1$	$\left(222\right)$
$\#1;2$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,422;2111$	$H_3,H_3,H_2,H_2,H_1,J_{11};H_1$	$\left(222\right)$
$\#1;3$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,322;2111$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(222\right)$
$\#1;4$	$211,211,311$	$J_0,J_0,K_1$	$31,31,31,31,311,322;2111$	$H_3,H_3,H_2,H_2,H_1,H_1;H_1$	$\left(222\right)$

Table 15. Table 15. Artin patterns with high complexity

Discriminant	${\tau }_1$	${\tau }_2$	${\tau }_4$
$-\mathrm{6\hspace{0.17em}328}$	$311,411,611$	$42,42,72,72,411,411;532$	$\left(622\right)$
$-\mathrm{7\hspace{0.17em}672}$	$311,311,311$	$311,311,321,321,422,422;21111$	$\left(22222\right)$
$-\mathrm{10\hspace{0.17em}803}$	$311,311,311$	$321,321,321,321,322,322;21111$	$\left(3321111\right)$
$-\mathrm{12\hspace{0.17em}595}$	$221,221,711$	$221,221,721,822,2211,3221;61111$	$\left(72221\right)$
$-\mathrm{19\hspace{0.17em}947}$	$221,221,411$	$221,221,421,522,2211,4411;31111$	$\left(43321\right)$
$-\mathrm{20\hspace{0.17em}155}$	$221,221,411$	$221,221,421,432,2211,3311;31111$	$\left(43332\right)$
$-\mathrm{27\hspace{0.17em}955}$	$221,221,411$	$522,522,2211,2211,2221,4311;31111$	$\left(5432211\right)$

Equations

$$H_1:=⟨x,J_0⟩,H_2:=⟨xy,J_0⟩,H_3:=⟨y,J_0⟩,\text{ with 1st layer Artin pattern }({\varkappa }_1,{\tau }_1).$$

$$H_1=⟨x,y^2,G^{\prime }⟩,H_2=⟨xy,y^2,G^{\prime }⟩,H_3=⟨y,x^2,G^{\prime }⟩,$$

$$K_1:=⟨x^2,G^{\prime }⟩,K_2:=⟨x^2{y}^2,G^{\prime }⟩,K_3:=⟨y^2,G^{\prime }⟩,\text{ with 3rd layer Artin pattern }({\varkappa }_3,{\tau }_3).$$

$$d_1\le d_2\le d_1+r+\theta ,$$

$$\theta =\{\begin{array}{ll}1& \text{if }K\text{ contains a primitive }p\text{-th root of unity},\\ 0& \text{otherwise}.\end{array}$$

$$2\le d_2\le 3+r,$$

$$r=\{\begin{array}{ll}1& \text{for real }K\text{ with }(r_1,r_2)=(2,0),\\ 0& \text{for imaginary }K\text{ with }(r_1,r_2)=(0,1).\end{array}$$

$$⟨128,31⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$$⟨256,285\text{ or }286⟩\to ⟨128,35⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$$\#1;2\to ⟨512,1605⟩\to ⟨256,284⟩\to ⟨128,35⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$$⟨512,1598⟩\to ⟨256,276⟩\to ⟨128,33⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$${\tau }_0$$

$${\tau }_2$$

$$-6052$$

$$-29444$$

$${\tau }_0$$

$${\tau }_2$$

$$-8103$$

$$-12104$$

$$-13727$$

$$-17751$$

$$-28067$$

$${\tau }_0$$

$${\tau }_2$$

$${\tau }_0$$

$${\tau }_2$$

$${\tau }_0$$

$${\tau }_2$$

$$\#2;140\stackrel{2}{\to }⟨512,227⟩\stackrel{2}{\to }⟨128,26⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$$\#1;4\to ⟨512,231⟩\stackrel{2}{\to }⟨128,26⟩\stackrel{2}{\to }⟨32,2⟩\to ⟨16,2⟩$$

$${\tau }_0$$

$${\tau }_2$$

$$-25144$$

$${\tau }_0$$

$${\tau }_2$$

$$M=⟨x,y\mid x^4=[t_3,y,y],y^4=[t_3,y,xy],[x^2,y]=1,w=1⟩,$$

$$G_{000}/{\gamma }_6{G}_{000}$$

$$G_{001}/{\gamma }_6{G}_{001}$$

$$G_{010}/{\gamma }_6{G}_{010}$$

Full text

Research supported by the Austrian Science Fund (FWF): project P 26008-N25

1. Introduction

Let $K$ be a quadratic number field with $2$-class group $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, that is, with abelian type invariants $(4,4)$. The strategy of pattern recognition via Artin transfers consists of three steps.

First, the number theoretic Artin pattern $\mathrm{AP}(K)$ of $K$ is defined as the collection $(\varkappa(K),\tau(K))$ of all kernels $\varkappa(K)=(\ker(T_{K,N}))_{N}$ and targets $\tau(K)=(\mathrm{Cl}_{2}(N))_{N}$ of class extension homomorphisms $T_{K,N}:\,\mathrm{Cl}_{2}(K)\to\mathrm{Cl}_{2}(N)$ from $K$ to unramified abelian extensions $K<N<\mathrm{F}_{2}^{1}(K)$ of $K$ within the Hilbert $2$-class field $\mathrm{F}_{2}^{1}(K)$ of $K$ [11]. It is determined by means of the class field theoretic routines of the computational algebra system Magma [5, 6, 8].

Second, the number theoretic Artin pattern $\mathrm{AP}(K)$ of $K$ is interpreted as the group theoretic Artin pattern $\mathrm{AP}(G)$ of the metabelian Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the second Hilbert $2$-class field $\mathrm{F}_{2}^{2}(K)$ of $K$, which consists of all kernels $\varkappa(G)=(\ker(T_{G,S}))_{S}$ and targets $\tau(G)=(S/S^{\prime})_{S}$ of Artin transfer homomorphisms $T_{G,S}:\,G/G^{\prime}\to S/S^{\prime}$ from $G$ to normal subgroups $G^{\prime}<S<G$ containing the commutator subgroup $G^{\prime}$ of $G$ [10], which is isomorphic to the $2$-class group $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))\simeq\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/\mathrm{F}_{2}^{1}(K))$ of the Hilbert $2$-class field $\mathrm{F}_{2}^{1}(K)$ of $K$. If the unramified abelian extension $N/K$ is the fixed field $N=\mathrm{Fix}(S)$ of the subgroup $S<G$, in the sense of the Galois correspondence, then $S/S^{\prime}\simeq\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/N)/\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/\mathrm{F}_{2}^{1}(N))\simeq\mathrm{Cl}_{2}(N)$, according to the Artin reciprocity law [1], and $\ker(T_{G,S})\simeq\ker(T_{K,N})$, by the theory of the Artin transfer [2].

Third, the group theoretic Artin pattern $\mathrm{AP}(G)$ is used as a search pattern in some pool of metabelian $2$-groups $G$ with abelianization $G/G^{\prime}\simeq C_{4}\times C_{4}$, for instance the Hall-Senior classification or the SmallGroups library [4], which contains all finite $2$-groups up to order $2^{9}$, or an extended pool, constructed selectively with the aid of the $p$-group generation algorithm by Newman [15] and O’Brien [16]. Frequently, this sifting process yields a unique candidate for the second $2$-class group $G=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the quadratic field $K$ [9], and the relation rank $d_{2}(G)$ of $G$ admits a decision about the length $\ell_{2}(K)$ of the $2$-class field tower of $K$, according to the Shafarevich cohomology criterion [14, §§ 1.2–1.3, pp. 75–76] in § 3.

Concerning the layout of this paper: The lattice of normal subgroups of $G/G^{\prime}\simeq C_{4}\times C_{4}$ is given as a foundation in § 2, imaginary base fields are classified in § 4, real base fields in § 5.

An extensive class of imaginary quadratic fields, presented in Theorems 4.2, 4.3 and 4.4, gives the opportunity to endow suitable subsets of the amorphous pool of finite $2$-groups with the structure of descendant trees by parent-child relations [13]. The periodicity of the branches of such trees with fixed coclass admits the characterization of infinitely many groups by a parametrized system of invariants in § 4.2, and the utilization of the fundamental principles of monotony, stabilization, polarization and mainline in the proof of many theorems.

2. Normal lattice of $G/G^{\prime}\simeq C_{4}\times C_{4}$

Let $G$ be a finite $2$-group with two generators $x,y$ such that $G=\langle x,y\rangle$ and $G/G^{\prime}\simeq C_{4}\times C_{4}$, i.e., $x^{4},y^{4}\in G^{\prime}$, where $G^{\prime}$ denotes the derived subgroup of $G$. Let $\tau_{0}=(G/G^{\prime})$ (0th layer). In the following description of the normal lattice of $G$, we use a notation similar to [3, § 2].

The Frattini subgroup $J_{0}:=\Phi(G)=\langle x^{2},y^{2},G^{\prime}\rangle$ of $G$ is the intersection of the three maximal subgroups

[TABLE]

They are of index $2$ in $G$ and can also be viewed as

[TABLE]

which shows that $H_{i}/G^{\prime}\simeq C_{4}\times C_{2}$, for each $1\leq i\leq 3$, whereas $J_{0}/G^{\prime}\simeq C_{2}\times C_{2}$.

In addition to $J_{0}$, there are six subgroups $J_{ik}$ with $1\leq i\leq 3$, $1\leq k\leq 2$, of index $4$ in $G$:

two subgroups of $H_{1}$, $\quad J_{11}:=\langle x,G^{\prime}\rangle,\quad J_{12}:=\langle xy^{2},G^{\prime}\rangle$,

two subgroups of $H_{2}$, $\quad J_{21}:=\langle xy,G^{\prime}\rangle,\quad J_{22}:=\langle xy^{3},G^{\prime}\rangle$,  and

two subgroups of $H_{3}$, $\quad J_{31}:=\langle y,G^{\prime}\rangle,\quad J_{32}:=\langle x^{2}y,G^{\prime}\rangle,\quad\text{ with 2nd layer Artin pattern }(\varkappa_{2},\tau_{2})$.

They all have a cyclic quotient $J_{ik}/G^{\prime}\simeq C_{4}$.

Finally, there are three subgroups of index $8$ in $G$:

[TABLE]

They have quotient $K_{i}/G^{\prime}\simeq C_{2}$, for each $1\leq i\leq 3$. Additionally, let $\tau_{4}=(G^{\prime}/G^{\prime\prime})$ (4th layer).

Logarithmic abelian type invariants are used for the components of $\tau$, for instance, $(4111)$ or even $(41^{3})$, with formal exponents indicating iteration, instead of $(16,2,2,2)$.

Among the remaining members of the normal lattice of $G$, we mention the terms $\gamma_{j}$ of the lower central series with $\gamma_{1}:=G$, and $\gamma_{j}:=[\gamma_{j-1},G]$, for $2\leq j\leq c$, and the terms $\zeta_{j}$ of the upper central series with $\zeta_{0}:=1$, and $\zeta_{j}/\zeta_{j-1}:=\mathrm{center}(G/\zeta_{j-1})$, for $1\leq j\leq c-1$. In both cases, $c=\mathrm{cl}(G)$ denotes the nilpotency class of $G$ such that $\gamma_{c+1}=1$ and $\zeta_{c}=G$. If $G$ is of order $2^{n}$, then $r=\mathrm{cc}(G)=n-c$ is the coclass of $G$.

Figure 1 shows the three parts of the normal lattice of $G$ above the derived subgroup $G^{\prime}$.

3. Shafarevich cohomology criterion

Let $p$ be a prime number and $G$ be a pro-$p$ group. Then the finite field $\mathbb{F}_{p}$ is a trivial $G$-module, and two crucial cohomological invariants of $G$ are the generator rank $d_{1}:=d_{p,1}(G):=\dim_{\mathbb{F}_{p}}H^{1}(G,\mathbb{F}_{p})$ and the relation rank $d_{2}:=d_{p,2}(G):=\dim_{\mathbb{F}_{p}}H^{2}(G,\mathbb{F}_{p})$. Shafarevich has established a necessary criterion for the Galois group $G=\mathrm{Gal}(\mathrm{F}_{p}^{\infty}(K)/K)$ of the Hilbert $p$-class field tower $\mathrm{F}_{p}^{\infty}(K)$ of a number field $K$ in terms of the generator rank and relation rank of $G$ [12, Thm. 5.1, p. 28]:

[TABLE]

where $r$ and $\theta$ are defined in the following way: if $(r_{1},r_{2})$ denotes the signature of $K$, then the torsion free Dirichlet unit rank of $K$ is given by $r=r_{1}+r_{2}-1$ and

[TABLE]

Applied to the particular case of $p=2$ and the $2$-class field tower group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of a quadratic number field $K$ with $2$-class group $G/G^{\prime}\simeq\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, and thus with Frattini quotient $G/\Phi(G)\simeq C_{2}\times C_{2}$, the Shafarevich criterion becomes

[TABLE]

since the generator rank is $d_{1}=2$, according to the Burnside basis theorem, $\theta=1$ trivially, and the Dirichlet unit rank is

[TABLE]

4. Imaginary quadratic fields of type $(4,4)$

According to Benjamin and Snyder [3, § 5, p. 1183], the discriminant of an imaginary quadratic field $K$ with $2$-class group $\mathrm{Cl}_{2}(K)$ of type $(4,4)$ is the product $d_{K}=d_{1}d_{2}d_{3}$ of three distinct prime discriminants $d_{1}<0$ and $d_{2},d_{3}>0$, and the three unramified quadratic extensions $N_{i}:=\mathrm{Fix}(H_{i})$ of $K$ are given by $N_{i}=K(\sqrt{d_{i}})$, for $1\leq i\leq 3$, such that generally $\#\mathrm{Cl}_{2}(N_{i})>16=\#\mathrm{Cl}_{2}(K)$ for $i\in\{2,3\}$ [3, Lem. 5.1]. The remaining extension $N_{1}$ is responsible for the $2$-rank of $\tau_{4}=\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$. In § 4.1, we first consider the case that $\#\mathrm{Cl}_{2}(N_{1})=16$ and $\#\mathrm{Cl}_{2}(N_{1k})=16$ for the two unramified cyclic quartic extensions $N_{1k}:=\mathrm{Fix}(J_{1k})$, $k\in\{1,2\}$, of $K$ which contain $N_{1}$, whence $\tau_{4}=\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ has rank $2$ by [3, Cor. 3.2, p. 1182].

4.1. Metabelian towers of Hilbert $2$-class fields

In Table 1, some discriminants $0>d_{K}>-10^{5}$ of imaginary quadratic fields $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ are shown together with the factorization of $d_{K}$ into three prime discriminants and the Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of the metabelian $2$-class field tower of $K$ in the notation of the SmallGroups library [4] and ANUPQ package [7].

The root paths of the occurring metabelian $2$-groups with orders up to $1024$ are given by (4.1).

[TABLE]

In accordance with the Shafarevich criterion, all these groups have relation rank $d_{2}=3$.

Theorem 4.1**.**

(Sporadic situation.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the sporadic group $\langle 128,31\rangle$ with coclass $4$.

Example 4.1**.**

The information in Formula (4.2) occurs rather sparsely as Artin pattern of imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$. The absolutely smallest two discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 4.1) Coarse selection using $\tau_{0}=(22)$ and the Artin pattern $\tau_{1}=(211,221,311)$ of the first layer only yields four candidates of order $128=2^{7}$ in [4]. See Table 2.

However, the unique candidate with correct invariants $\varkappa_{1}$, $\tau_{2}$, $\varkappa_{2}$ is $\langle 128,31\rangle$. All candidates share the common invariant $\tau_{4}=(21)$. So the second layer enables the fine selection.

Candidates of bigger orders $256=2^{8}$ and $512=2^{9}$ possess too big components of $\tau_{2}$ and the search can be stopped, according to the monotony principle [14, § 1.7, Thm. 1.21, p. 79].

Here we have $\mathrm{Cl}_{2}(N_{1})=\mathrm{Cl}_{2}(N_{1k})=(211)$, $k\in\{1,2\}$, and indeed $\tau_{4}=(21)$ [3, Cor. 3.2]. ∎

Theorem 4.2**.**

(Ground state.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the periodic group $\langle 256,285\rangle$ (see § 4.2) with coclass $4$.

Example 4.2**.**

The information in Formula (4.4) seems to be the most frequent Artin pattern of imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$. The absolutely smallest ten discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 4.2) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$, $\tau_{1}=(211,221,411)$ and $\tau_{2}=(22,22,31,31,41,41;321)$ yields $4$ groups $\langle 256,i\rangle$ with identifiers $284\leq i\leq 287$. Only $\langle 256,285\rangle$ possesses correct capitulation type $\varkappa_{1}=(J_{0},K_{1},K_{3})$ and $\varkappa_{2}=(H_{1},H_{1},H_{2},H_{2},H_{3},H_{3};J_{0})$ (Table 5). Here, $\mathrm{Cl}_{2}(N_{1})=(211)$, $\mathrm{Cl}_{2}(N_{1k})=(31)$, $\tau_{4}=(31)$ [3, Cor. 3.2] ∎

Theorem 4.3**.**

(Ground state variant.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the periodic group $\langle 256,286\rangle$ (see § 4.2) with coclass $4$.

Example 4.3**.**

Only a single imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$ and Artin pattern in Formula (4.6) is known up to now. The discriminant $d_{K}$ of this field is $-11\,972=(-4)\cdot 41\cdot 73$.

Proof.

(Proof of Theorem 4.3) As in the previous proof, searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$, $\tau_{1}=(211,221,411)$ and $\tau_{2}=(22,22,31,31,41,41;321)$ yields $4$ groups $\langle 256,i\rangle$ with $284\leq i\leq 287$. However, only $\langle 256,286\rangle$ possesses the correct capitulation type $\varkappa_{1}=(J_{0},K_{1},K_{1})$ with two equal kernels and $\varkappa_{2}=(H_{1},H_{1},H_{2},H_{2},H_{1},H_{1};H_{1})$. See Table 5. ∎

Theorem 4.4**.**

(Excited state.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the periodic group $\langle 512,1605\rangle-\#1;2$ (see § 4.2) with order $1024$ and coclass $4$.

Example 4.4**.**

Only a single imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$ and Artin pattern in Formula (4.7) is known up to now. The discriminant $d_{K}$ of this field is $-14\,547=(-3)\cdot 13\cdot 373$.

Proof.

(Proof of Theorem 4.4) None of the $2$-groups in the SmallGroups library [4] hits the pattern up to the first layer $\tau_{0}=(22)$, $\tau_{1}=(211,221,611)$, let alone the complete pattern up to the second layer $\tau_{2}=(22,22,31,31,61,61;521)$. However, the capitulation types $\varkappa_{1}$ and $\varkappa_{2}$ in Formula (4.7) and Formula (4.4) coincide precisely, whereas the abelian quotient invariants in these formulas coincide only partially, while some components show an increase of $\tau_{1}=(211,221,411)$ and $\tau_{2}=(22,22,31,31,41,41;321)$. As explained in section 4.2, these observations suggest that the Artin pattern (4.7) is an excited state of the ground state (4.4). Hence, $\langle 256,284\rangle$ is selected as starting point for extending the pool by $2$-groups with orders bigger than $512$, and thus outside of the SmallGroups database. Descendants are generated with depth two (including grand children) but only step size $1$ (since bifurcations to coclass higher than $4$ cannot occur). Among the $18$ groups generated in this manner, there are $4$ hits $\langle 512,1605\rangle-\#1;i$ with $1\leq i\leq 4$ of the complete pattern $\tau_{0}$, $\tau_{1}$ and $\tau_{2}$. But only $\langle 512,1605\rangle-\#1;2$ has correct capitulation type (Table 5). ∎

Theorem 4.5**.**

(Another tree.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the periodic group $\langle 512,1598\rangle$ with centre $\zeta\simeq C_{4}\times C_{2}$ on the tree $\mathcal{T}^{4}(\langle 128,33\rangle)$.

Example 4.5**.**

Only a single imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$ and Artin pattern in Formula (4.8) is known up to now. The discriminant $d_{K}$ of this field is $-17\,476=(-4)\cdot 17\cdot 257$.

Proof.

(Proof of Theorem 4.5) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$, $\tau_{1}=(211,311,511)$ and $\tau_{2}=(31,31,31,31,51,51;421)$ yields $4$ groups $\langle 512,i\rangle$ with identifiers $1595\leq i\leq 1598$. But only $\langle 512,1598\rangle$ possesses the correct capitulation type $\varkappa_{1}=(J_{0},K_{3},K_{3})$ with two equal kernels and $\varkappa_{2}=(H_{2},H_{2},H_{3},H_{3},H_{2},H_{2};H_{2})$. See Table 3. Here, we have $\mathrm{Cl}_{2}(N_{1})=(211)$, $\mathrm{Cl}_{2}(N_{1k})=(31)$, for $k\in\{1,2\}$, and indeed $\tau_{4}=(41)$ [3, Cor. 3.2]. ∎

4.2. The first descendant tree of $2$-groups with type $(4,4)$

The first descendant tree $\mathcal{T}^{4}(R)$ of finite $2$-groups $G$ with abelian type invariants $(4,4)$ and fixed coclass $\mathrm{cc}(G)=4$ was discovered in May 2019 by analyzing the second $2$-class group $\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of imaginary quadratic fields $K$ with discriminants $d_{K}\in\{-8\,103,\ -11\,972,\ -14\,547\}$ and $2$-class group $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$. Its metabelian root $R=\langle 128,35\rangle$ is connected with the general abelian root $A=\langle 16,2\rangle\simeq C_{4}\times C_{4}$ by the path $R\stackrel{{\scriptstyle 2}}{{\to}}\langle 32,2\rangle\to A$. Up to order $1024$, the tree is drawn in Figure 2.

The mainline is characterized graph theoretically by the descendant numbers $(N,C)=(8,2)$. It is possible to partition the eight immediate descendants of a mainline vertex group theoretically: Four vertices ($\bigcirc$) possess a centre $\zeta\simeq C_{4}\times C_{2}$, one of them belongs to the mainline, the other three are terminal. The other four vertices $(\bullet$) have $\zeta\simeq C_{2}\times C_{2}$, the third of them has nuclear rank $\nu=1$ and descendant numbers $(N,C)=(2,0)$, the other three are leaves. Both vertices of depth two ($\square$) have $\zeta\simeq C_{2}$. (By $\zeta=\zeta_{1}$ we denote the first term of the upper central series.)

The tree is an excellent paradigm for the principle of stabilization and polarization of certain components of the Artin pattern. We avoid complications by restricting ourselves to the vertices ($\bigcirc$) with centre $\zeta\simeq C_{4}\times C_{2}$ which are located on and left from the mainline in Figure 2.

Among the transfer targets (abelian quotient invariants), the first two components of the first layer $\tau_{1}=(211,221,\ast)$ and the first four components of the second layer $\tau_{2}=(22,22,31,31,\ast,\ast;\ast)$ remain stable on the entire tree. Similarly for the transfer kernels (capitulation kernels), the first two components of the first layer $\varkappa_{1}=(J_{0},K_{1},\ast)$ and the first four components of the second layer $\varkappa_{2}=(H_{1},H_{1},H_{2},H_{2},\ast,\ast;\ast)$ remain stable.

The polarization appears in a different manner for targets and kernels. The polarized components of the transfer targets become strictly bigger from parent to descendant (according to their isotonic monotony principle), i.e., with increasing order. Starting at the root, we successively have $\tau_{1}(3)\in\{311,411,511,611,\ldots\}$, $\tau_{2}(5)=\tau_{2}(6)\in\{31,41,51,61,\ldots\}$, $\tau_{2}(7)\in\{221,321,421,521,\ldots\}$, and $\tau_{4}\in\{21,31,41,51,\ldots\}$. This phenomenon gives rise to countably many states, corresponding to the infinitely many branches of the tree in vertical direction (which are all isomorphic as graphs for this tree with period length $1$), and indicated in Table 4.

On the other hand, the polarized components of the transfer kernels (which may become smaller from parent to descendant, according to their antitonic monotony principle) vary in the same way for each fixed order $\varkappa_{1}(3)\in\{J_{0},K_{3},K_{1},K_{2}\}$, $\varkappa_{2}(5)=\varkappa_{2}(6)\in\{G,H_{3},H_{1},H_{2}\}$, and $\varkappa_{2}(7)\in\{H_{1},J_{0},H_{1},J_{0}\}$, with respect to increasing SmallGroups identifiers. Hence, there are only finitely many variants within a fixed state in horizontal direction, characterized by distinct capitulation types in Table 5.

The tree is also an excellent paradigm for the mainline principle, which states that mainline vertices have bigger capitulation kernels in their polarization. Indeed, we have $J_{0}>K_{i}$ for $\varkappa_{1}(3)$, $G>H_{i}$ for $\varkappa_{2}(5)=\varkappa_{2}(6)$, and $H_{1}>J_{0}$ for $\varkappa_{2}(7)$.

4.3. Non-metabelian towers of Hilbert $2$-class fields

Table 6 shows some discriminants $0>d_{K}>-10^{5}$ of imaginary quadratic fields $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, their factorization into three prime discriminants and the metabelianization $G/G^{\prime\prime}\simeq\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of the non-metabelian $2$-class field tower of $K$ in the notation of the SmallGroups library [4] and ANUPQ package [7].

The root path of the occurring metabelian $2$-groups with orders $2048$ and $1024$ is given by (4.9).

[TABLE]

In accordance with the Shafarevich criterion, these metabelianizations have relation rank $d_{2}(G/G^{\prime\prime})=4$, and therefore cannot be $2$-tower groups of imaginary quadratic fields. The coclass is $6$, resp. $5$.

Theorem 4.6**.**

(Three-stage tower.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

The $2$-class tower of $K$ has length $\ell_{2}(K)=3$ and the metabelianization $G/G^{\prime\prime}=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ is isomorphic to $\langle 512,231\rangle-\#1;4$ with order $1024$. The group $G$ itself is isomorphic to $\langle 512,231\rangle-\#2;j$ with some $j\in\{36,39,52,55\}$ and order $2048$.

Example 4.6**.**

There are only two known discriminants $0>d_{K}>-10^{5}$ of imaginary quadratic fields $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ and Artin pattern in Formula (4.10):

[TABLE]

Proof.

(Proof of Theorem 4.6) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$ and $\tau_{1}=(211,221,411)$, up to the first layer alone, yields $8$ groups $\langle 256,i\rangle$ with $246\leq i\leq 249$ and $284\leq i\leq 287$ and $28$ groups $\langle 512,j\rangle$ with $227\leq j\leq 242$, $1539\leq j\leq 1546$ and $1609\leq j\leq 1612$. But none of the groups in the SmallGroups library hits the complete pattern up to the second layer, including $\tau_{2}=(22,22,41,41,222,2111;3111)$. Some of the $36$ candidates have too big components in their second layer pattern $\tau_{2}$: The four groups $\langle 256,i\rangle$ with $284\leq i\leq 287$ have inadequate Frattini invariants $(321)$ instead of $(3111)$, the eight groups $\langle 512,j\rangle$ with $1539\leq j\leq 1542$ and $1609\leq j\leq 1612$ have two inadequate invariants $(51)$ instead of $(41)$, and they can be eliminated as parents for extending the pool by the $p$-group generation algorithm [15, 16], according to the monotony principle [14, § 1.7, Thm. 1.21, p. 79]. The four groups $\langle 512,j\rangle$ with $1543\leq j\leq 1546$ can be eliminated, because they are terminal (without descendants).

Since all the other groups share $\langle 128,26\rangle$ as their common parent, this group is chosen as starting point for selectively constructing an extended pool of $2$-groups with orders bigger than $512$, and thus outside of the SmallGroups database. Descendants are generated with depth two (including grand children) and step sizes $1$ and $2$ (including bifurcations to coclass higher than $4$). Among the $1754$ groups generated in this manner, there are $176$ hits of the complete pattern $\tau_{0}=(22)$, $\tau_{1}=(211,221,411)$, $\tau_{2}=(22,22,41,41,222,2111;3111)$ and $\tau_{4}=(321)$ of abelian type invariants, but without considering any capitulation patterns $\varkappa_{1}$ and $\varkappa_{2}$. $16$ of the hits are metabelian with order $1024$ and coclass $5$, and can be partitioned into four batches with $4$ members each. The remaining $160$ hits are non-metabelian with derived length $3$, order $2048$ and coclass $6$, and can be partitioned into four batches, one with $64$ members and three with $32$ members each. There is a one-to-one correspondence between metabelian and non-metabelian batches, where the members of the former are metabelianizations of the latter, and furthermore both share a common parent. The pair of batches with parent $\langle 512,227\rangle$ is eliminated, due to an inadequate capitulation type $\varkappa_{1}=(J_{0},K_{3},J_{0})$. (For this pair of batches, $16$ non-metabelian groups map to the same metabelianization, for the other pairs of batches, only $8$ map to the same.) Similarly, the pair of batches with parent $\langle 512,228\rangle$ is eliminated, due to an inadequate capitulation type $\varkappa_{1}=(J_{0},K_{3},K_{3})$. The two pairs of batches with parents $\langle 512,231\rangle$, resp. $\langle 512,232\rangle$, have admissible $\varkappa_{1}=(J_{0},K_{3},K_{1})$, resp. $\varkappa_{1}=(J_{0},K_{3},K_{2})$. For the final decision, the second layer capitulation type $\varkappa_{2}$ must be taken into acount. The pair of batches with parent $\langle 512,232\rangle$ is eliminated, because $\varkappa_{2}=(H_{3},H_{3},H_{2},H_{2},\ast,H_{1};J_{0})$ assigns different capitulation kernels to the components with abelian type invariants $(41)$ and $(2111)$. For the last remaining pair of batches with parent $\langle 512,231\rangle$, Table 7 shows the development of the Artin pattern along the root path, for each of the four members $\#1;i$ with $i\in\{3,4,7,8\}$ of the metabelian batch. They all share the common stabilization $\varkappa_{2}=(H_{3},H_{3},H_{1},H_{1},\ast,H_{1};J_{0})$, but only $\#1;4$ possesses the correct polarization with $\ast$ replaced by $H_{2}$ such that the component with abelian type invariants $(222)$ has a capitulation kernel different from all other kernels.

Among the eight members of the associated non-metabelian batch, four are capable with relation rank $d_{2}=4$, and only the other four siblings $\#2;j$ with $j\in\{36,39,52,55\}$ which are terminal with required relation rank $d_{2}=3$ are candidates for the $2$-tower group $G$ with three stages.

Here, we have $\mathrm{Cl}_{2}(N_{1})=(211)$, but $\mathrm{Cl}_{2}(N_{11})=(222)$, $\mathrm{Cl}_{2}(N_{12})=(2111)$ with order bigger than $16$, and thus $\tau_{4}=(321)$ with rank three [3, Cor. 3.2]. ∎

Theorem 4.7**.**

(Tower with $\ell_{2}\geq 3$.) Let $K$ be an imaginary quadratic field with Artin pattern

[TABLE]

The $2$-class tower of $K$ has length $\ell_{2}(K)\geq 3$ and the metabelianization $G/G^{\prime\prime}=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ is isomorphic to $\langle 512,227\rangle-\#2;140$ of order $2048$ and coclass $6$. The smallest possibility for the $2$-tower group $G$ itself is an isomorphism to one of the four non-metabelian groups $\langle 512,227\rangle-\#3;j$ with $j\in\{39,62,231,254\}$ (see § 4.4) of order $4096$, coclass $7$ and derived length $3$.

Example 4.7**.**

$-6\,123$ is the single known discriminant $0>d_{K}>-10^{5}$ of an imaginary quadratic field $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ and Artin pattern in Formula (4.12). It is mentioned in [3, Exm. 7.2, p. 1192] without explicit invariant $\tau_{4}=(421)$. However, $\ell_{2}(K)$ and $G$ are unknown (§ 4.4).

Proof.

(Proof of Theorem 4.7) We have intentionally delayed this proof, because we shall need our experience from the proof of Theorem 4.6. Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$ and $\tau_{1}=(211,221,511)$, up to the first layer alone, yields $8$ groups $\langle 512,j\rangle$ with $1535\leq j\leq 1538$ and parent $\langle 256,246\rangle$, resp. $1605\leq j\leq 1608$ and parent $\langle 256,284\rangle$. But none of the groups in the SmallGroups library hits the complete pattern up to the second layer, including $\tau_{2}=(22,22,51,51,222,2111;4111)$. Construction of descendants with starting point $\langle 256,246\rangle$ or $\langle 256,284\rangle$ can be terminated after depth one already, according to the monotony principle [14, § 1.7, Thm. 1.21, p. 79], since certain components of $\tau_{2}$ become too big. At this point we lack any further traces.

Guided by the proof of Theorem 4.6, we remember the promising starting group $\langle 128,26\rangle$ with $\tau_{1}=(211,221,311)$ and $\tau_{2}=(22,22,31,31,211,211;2111)$, which would be compatible with our search pattern (4.10), according to the monotony principle. Indeed, a similar construction of $1754$ descendants as in the proof of Theorem 4.6 yields $16$ hits by metabelian groups of order $2048$ with common parent $\langle 512,227\rangle$. Only one of them, namely $\langle 512,227\rangle-\#2;140$ possesses the required capitulation type with $\varkappa_{1}=(J_{0},K_{1},K_{3})$ and $\varkappa_{2}=(H_{1},H_{1},H_{2},H_{2},H_{3},H_{2};J_{0})$.

Finally, we construct the $1016$ immediate descendants of all step sizes $1,2,3$ of $\langle 512,227\rangle$ (including the trifurcation to coclass $7$) and find eight non-metabelian groups with metabelianization $\langle 512,227\rangle-\#2;140$. Four of them are capable with relation rank $d_{2}=4$, and only the other four siblings $\#3;j$ with $j\in\{39,62,231,254\}$ which are terminal with required relation rank $d_{2}=3$ are candidates for the $2$-tower group $G$ with three stages.

Here, we have $\mathrm{Cl}_{2}(N_{1})=(211)$, but $\mathrm{Cl}_{2}(N_{11})=(222)$, $\mathrm{Cl}_{2}(N_{12})=(2111)$ with order bigger than $16$, and thus $\tau_{4}=(421)$ with rank three [3, Cor. 3.2]. ∎

The situations in Theorem 4.6 resp. Theorem 4.7 were called advanced tree topologies of type fork and siblings in [14, § 5, pp. 89–92]. The fork is $\langle 512,231\rangle$ resp. $\langle 512,227\rangle$.

4.4. Limits and higher abelian quotient invariants

Imaginary quadratic fields $K$ with Artin pattern in Formula (4.12), in particular the field with discriminant $d_{K}=-6\,123$ in Example 4.7, give rise to problems with determining the exact length $\ell_{2}(K)\geq 3$ of their $2$-class field tower. The following theorem illuminates the minimal candidates for the $2$-tower group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of such fields $K$ in Theorem 4.7 more closely from the viewpoint of finitely presented $2$-groups, and shows that the tower length $\ell_{2}(K)$ cannot be bounded by $3$.

Theorem 4.8**.**

*(Pro-$2$ groups as limit groups)

The eight non-metabelian $2$-groups of order $4096$ with metabelianization $M:=\langle 512,227\rangle-\#2;140$,*

[TABLE]

are the class-$5$ quotients

[TABLE]

of the following finitely presented groups:

[TABLE]

where $s_{2}:=[y,x]$ is the main commutator, $t_{3}:=[s_{2},y]$, and $w$ denotes the group word $w(x,y):=[t_{3},s_{2}]=[[y,x,y],[y,x]]$.

There exists a non-metabelian $2$-group $\mathfrak{G}_{20}$ of order $2^{20}$ with derived length $4$, relation rank $3$ and coclass $9$ with metabelianization $M$ and the following root path:

[TABLE]

This group $\mathfrak{G}_{20}$ is descendant $G_{100}/\gamma_{12}G_{100}$ of $G_{100}/\gamma_{6}G_{100}$ and class-$11$* quotient of $G_{100}$.*

However, in spite of Theorem 4.8, we cannot determine the $2$-tower group $G$ of the field $K$ with discriminant $d_{K}=-6\,123$, because its abelian quotient invariants $\tau_{2,1}$ of higher order in Table 8 do not match any known finite $2$-group.

4.5. Unexpected two-stage towers of Hilbert $2$-class fields

Table 9 shows a discriminant $0>d_{K}>-10^{5}$ of an imaginary quadratic field $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, its factorization into three prime discriminants and eight possible candidates for the metabelian Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of the $2$-class field tower of $K$ with length $\ell_{2}(K)=2$ in the notation of the SmallGroups library [4] and ANUPQ package [7].

The root paths of the occurring eight metabelian $2$-groups with order $2048$ are given by (4.17),

[TABLE]

with respect to the lower exponent-$2$ central series, and by (4.18),

[TABLE]

with respect to the (usual) lower central series. The groups are of coclass $6$.

Theorem 4.9**.**

(Tower with length two.) Let $K$ be imaginary quadratic with Artin pattern

[TABLE]

The $2$-class tower of $K$ has length $\ell_{2}(K)=2$ and the metabelian Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ is isomorphic to one of eight groups $\langle 256,28\rangle-\#2;i-\#1;j$ with order $2048$ and relation rank $d_{2}(G)=3$, where $i\in\{24,26\}$ and $1\leq j\leq 4$.

Here, we have $\mathrm{Cl}_{2}(N_{1})=(311)$ with order bigger than $16$ in the first layer already, whence $\tau_{4}$ has $2$-rank at least three [3, Cor. 3.2], independently of the $2$-class numbers $\#\mathrm{Cl}_{2}(N_{1k})$. Indeed, the precise structure is $\tau_{4}=(421)$, but nevertheless the tower length is only $\ell_{2}(K)=2$.

Example 4.8**.**

The information in Formula (4.19) occurs rather sparsely as Artin pattern of imaginary quadratic fields $K$ with $2$-class group of type $(4,4)$. The absolutely smallest three discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 4.9) In the SmallGroups database [4] the pattern $\tau_{0}=(22)$ and $\tau_{1}=(311,311,411)$, up to the first layer alone, sets in with order $512=2^{9}$. It is hit by $10$ groups $\langle 512,i\rangle$ with $798\leq i\leq 801$, $820\leq i\leq 822$ and $851\leq i\leq 853$. But none of the groups in the SmallGroups library hits the complete pattern up to the second layer, including $\tau_{2}=(32,32,52,52,411,411;322)$. Some of the $10$ candidates have too big components in their second layer pattern $\tau_{2}$: The three groups $\langle 512,i\rangle$ with $820\leq i\leq 822$ and common parent $\langle 256,32\rangle$ have inadequate Frattini invariants $(3211)$ instead of $(322)$, the three groups $\langle 512,i\rangle$ with $851\leq i\leq 853$and common parent $\langle 256,38\rangle$ have four inadequate invariants $(311)$ instead of $(32)$, and they can be eliminated as parents for extending the pool by the $p$-group generation algorithm [15, 16], according to the monotony principle [14, § 1.7, Thm. 1.21, p. 79]. The four groups $\langle 512,i\rangle$ with $798\leq i\leq 801$ and common parent $\langle 256,28\rangle$ possess compatible second layer $\tau_{2}=(32,32,42,42,311,311;322)$ but the descendants of the first two groups show stable components $(311)$ instead of $(411)$, and thus the further construction of descendants can be stopped.

Instead, we select the group $\langle 256,28\rangle$ as starting point for constructing an extended pool of $2$-groups with orders bigger than $512$, and thus outside of the SmallGroups database. Descendants are generated with depth three (including great grand children) and step sizes $1$, $2$ and $3$ (including bi- and trifurcations to coclass bigger than $5$). Among the $4178$ groups generated in this way, there are $8$ hits of the complete pattern $\tau_{0}=(22)$, $\tau_{1}=(311,311,411)$, $\tau_{2}=(32,32,52,52,411,411;322)$ and $\tau_{4}=(421)$ of abelian type invariants, but without considering any capitulation patterns $\varkappa_{1}$ and $\varkappa_{2}$. The eight groups are $\langle 256,28\rangle-\#2;i-\#1;j$ with order $2048$ and admissible relation rank $d_{2}=3$, where $i\in\{24,26\}$ and $1\leq j\leq 4$, and they all share the desired capitulation type $\varkappa_{1}=(J_{0},K_{2},K_{3})$ and $\varkappa_{2}=(H_{1},H_{1},J_{31},J_{32},J_{21},J_{22};J_{0})$. Here, the trifurcation does not lead to non-metabelian groups with Artin pattern (4.19), and the tower must necessarily be of length $\ell_{2}(K)=2$, since topologies of type parent — child or fork — sibling are discouraged. ∎

5. Real quadratic fields of type $(4,4)$

In § 4, we have seen that the root path of all $2$-class field tower groups $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of imaginary quadratic fields $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ contains at least one bifurcation, and thus $G$ must be at least of coclass $\mathrm{cc}(G)=4$. The deeper reason for this fact is a restriction for the order of the capitulation kernel $\ker(T_{K,N})$ of $K$ with respect to an unramified cyclic extension $N/K$, according to the Theorem on the Herbrand quotient of the unit group $U_{N}$ of $N$:

[TABLE]

For an imaginary field $K$, the unit norm index is bounded by $1\leq(U_{K}:\mathrm{N}_{N/K}(U_{N}))\leq 2$, and Formula (5.1) yields the estimate $4\leq\#\ker(T_{K,N})\leq 8$ for an extension of degree $[N:K]=4$, whence $\varkappa_{2}(K)$ cannot contain a total capitulation kernel. However, for the finite $2$-groups $G$ near the root $C_{4}\times C_{4}$, $\varkappa_{2}(G)$ always contains at least one total capitulation kernel, whence the imaginary tower group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ must be located rather far away from the root.

In the present section, we shall see that the root path of the $2$-class field tower group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of a real quadratic field $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ does not necessarily contain a bifurcation, and thus $G$ may be of the minimal possible coclass $\mathrm{cc}(G)=3$. For a real field $K$ and an extension of degree $[N:K]=4$, Formula (5.1) yields the less severe constraint $4\leq\#\ker(T_{K,N})\leq 16$, since $1\leq(U_{K}:\mathrm{N}_{N/K}(U_{N}))\leq 4$, due to the existence of the fundamental unit $\eta$ in $U_{K}$. In fact, the real tower group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ may even coincide with the root:

5.1. Abelian towers of Hilbert $2$-class fields

Theorem 5.1**.**

(Single-stage tower.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=1$ and Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{1}(K)/K)$ isomorphic to the abelian root $\langle 16,2\rangle\simeq C_{4}\times C_{4}$ with relation rank $d_{2}=3$.

Example 5.1**.**

The information in Formula (5.2) cannot occur for imaginary quadratic fields but seems to be a densely populated Artin pattern of real quadratic fields $K$ with $2$-class group of type $(4,4)$. The smallest ten discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 5.1) According to item (1) of [3, Cor. 3.1, p. 1182], the $2$-class field tower of $K$ is abelian with $\mathrm{F}_{2}^{\infty}(K)=\mathrm{F}_{2}^{1}(K)$ and $\#\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))=1$ (i.e., $\tau_{4}=(0)$) if and only if $\#\mathrm{Cl}_{2}(\mathrm{Fix}(H_{i}))=\frac{1}{2}\cdot\#\mathrm{Cl}_{2}(K)$, for $1\leq i\leq 3$ (which is true for $\tau_{1}=(21,21,21)$, since $8=\frac{1}{2}\cdot 16$). Pattern (5.2) is unique for the abelian group $\langle 16,2\rangle$ in the SmallGroups library [4]. ∎

5.2. Metabelian towers of Hilbert $2$-class fields

In Table 10, some discriminants $0<d_{K}<10^{6}$ of real quadratic fields $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ are shown together with the factorization of $d_{K}$ into prime discriminants and the Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of the metabelian $2$-class field tower of $K$ in the notation of the SmallGroups library [4] and ANUPQ package [7].

The root paths of the occurring metabelian $2$-groups with orders up to $512$ are given by (5.4).

[TABLE]

In accordance with the Shafarevich criterion, all these groups have relation rank $3\leq d_{2}\leq 4$.

Theorem 5.2**.**

(Sporadic situation.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the sporadic group $\langle 32,4\rangle$ with relation rank $d_{2}=3$.

Example 5.2**.**

The information in Formula (5.5) seems to be the most frequent Artin pattern of real quadratic fields $K$ with $2$-class group of type $(4,4)$. The smallest ten discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 5.2) According to item (2) of [3, Cor. 3.1, p. 1182], the $2$-class field tower of $K$ is metabelian with $\mathrm{F}_{2}^{\infty}(K)=\mathrm{F}_{2}^{2}(K)$ and $\#\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))=2$ (i.e., $\tau_{4}=(1)$) if and only if $\#\mathrm{Cl}_{2}(\mathrm{Fix}(J_{0}))=\frac{1}{2}\cdot\#\mathrm{Cl}_{2}(K)$ (which is true for $\tau_{2}=(3,3,3,3,21,21;21)$, since $8=\frac{1}{2}\cdot 16$). Pattern (5.5) is unique for $\langle 32,4\rangle$ in [4], whereas $\tau_{1}$ alone also appears for the child $\langle 64,28\rangle$. ∎

Theorem 5.3**.**

(Another sporadic situation 1.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the sporadic group $\langle 64,20\rangle$ with relation rank $d_{2}=4$.

Example 5.3**.**

The information in Formula (5.7) cannot occur for imaginary quadratic fields but seems to be a sparsely populated Artin pattern of real quadratic fields $K$ with $2$-class group of type $(4,4)$. The smallest four discriminants $d_{K}$ with this pattern are

[TABLE]

Note that all discriminants are divisible by four primes, whence this situation with $\tau_{4}=(2)$ lies outside of the scope of Theorems 6.2 and 6.3 in [3, pp. 1186–1191]

Proof.

(Proof of Theorem 5.3) According to item (3) of [3, Cor. 3.1, p. 1182], the $2$-class field tower of $K$ is metabelian with $\mathrm{F}_{2}^{\infty}(K)=\mathrm{F}_{2}^{2}(K)$ and $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ cyclic of order $\geq 4$ (i.e., $\tau_{4}=(e)$ with $e\geq 2$) if (but not only if) $\#\mathrm{Cl}_{2}(\mathrm{Fix}(H_{i}))=\#\mathrm{Cl}_{2}(K)$, for $1\leq i\leq 2$, $\#\mathrm{Cl}_{2}(\mathrm{Fix}(H_{3}))\geq 2\cdot\#\mathrm{Cl}_{2}(K)$, and $\#\mathrm{Cl}_{2}(\mathrm{Fix}(J_{ij}))=\frac{1}{2}\cdot\#\mathrm{Cl}_{2}(K)$, for $1\leq i,j\leq 2$ (which is true for $\tau_{1}=(211,211,221)$ and $\tau_{2}=(21,21,21,21,22,22;211)$, since $32=2\cdot 16$ and $8=\frac{1}{2}\cdot 16$). Pattern (5.7) is unique for $\langle 64,20\rangle$ in the SmallGroups database [4], whereas $\tau_{1}$ alone also appears for orders $128,256,512$. ∎

Theorem 5.4**.**

(Another sporadic situation 2.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the sporadic group $\langle 128,19\rangle$ with relation rank $d_{2}=3$.

Example 5.4**.**

The information in Formula (5.9) seems to be a sparsely populated Artin pattern of real quadratic fields $K$ with $2$-class group of type $(4,4)$. The smallest two discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 5.4) Pattern (5.9) is unique for $\langle 128,19\rangle$ in the SmallGroups library [4], whereas $\tau_{1}$ alone also appears for orders $64,256,512$ and other groups of order $128$. Moreover, the transfer targets $\tau_{0}$, $\tau_{1}$, $\tau_{2}$, $\tau_{4}$ also occur for $\langle 128,18\rangle$, which has, however, different tranfer kernels $\varkappa_{1}=(J_{0},J_{0},J_{0})$ and $\varkappa_{2}=(H_{1},H_{1},G,G,H_{3},H_{3};H_{2})$. ∎

Theorem 5.5**.**

(Coclass tree.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to the periodic group $\langle 256,281\rangle$ with $\zeta\simeq C_{2}\times C_{2}$ and $d_{2}=3$ on the tree $\mathcal{T}^{4}(\langle 128,33\rangle)$.

Example 5.5**.**

The information in Formula (5.11) seems to be a sparsely populated Artin pattern of real quadratic fields $K$ with $2$-class group of type $(4,4)$. The smallest two discriminants $d_{K}$ with this pattern are

[TABLE]

Proof.

(Proof of Theorem 5.5) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$, $\tau_{1}=(211,311,311)$ and $\tau_{2}=(31,31,31,31,41,41;321)$ yields $4$ groups $\langle 256,i\rangle$ with identifiers $280\leq i\leq 283$ and $2$ groups $\langle 512,j\rangle$ with identifiers $1603\leq j\leq 1604$. But only $\langle 256,281\rangle$ possesses the correct capitulation type $\varkappa_{1}=(J_{0},K_{2},J_{0})$ and $\varkappa_{2}=(H_{1},H_{1},H_{2},H_{2},G,G;J_{0})$. See Table 11. Here, we have $\mathrm{Cl}_{2}(N_{1})=(211)$, $\mathrm{Cl}_{2}(N_{1k})=(31)$, for $k\in\{1,2\}$, and indeed $\tau_{4}=(31)$ with $2$-rank two, according to [3, Cor. 3.2]. ∎

Theorem 5.6**.**

(Another tree.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class field tower of $K$ has length $\ell_{2}(K)=2$ and metabelian Galois group $\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ isomorphic to either $\langle 256,210\rangle$ or $\langle 256,211\rangle$ with $d_{2}=3$ on the tree $\mathcal{T}^{4}(\langle 128,18\rangle)$.

Example 5.6**.**

Only a single real quadratic fields $K$ with $2$-class group of type $(4,4)$ and Artin pattern in Formula (5.13) is known up to now. The discriminant $d_{K}$ of this field is $26\,245=5\cdot 29\cdot 181$.

Proof.

(Proof of Theorem 5.6) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$, $\tau_{1}=(211,211,311)$ and $\tau_{2}=(31,31,22,22,32,32;321)$ yields $4$ groups $\langle 256,i\rangle$ with identifiers $209\leq i\leq 212$. But only $\langle 256,210\rangle$ and $\langle 256,211\rangle$ possess the correct capitulation type $\varkappa_{1}=(J_{0},J_{0},J_{0})$ and $\varkappa_{2}=(H_{1},H_{1},G,G,H_{3},H_{3};H_{2})$. See Table 12. ∎

5.3. Unexpected towers of Hilbert $2$-class fields with $\ell_{2}(K)=2$

Table 13 shows a discriminant $0<d_{K}<10^{6}$ of a real quadratic field $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, its factorization into prime discriminants and the metabelian Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ of the $2$-class field tower of $K$ with unexpected length $\ell_{2}(K)=2$ in the notation of the SmallGroups library [4] and ANUPQ package [7].

The root path of the occurring metabelian $2$-group with order $1024$ is given by (5.14).

[TABLE]

Here, the Shafarevich criterion cannot be used for a decision about the length, since this metabelian group has relation rank $d_{2}=3$, and therefore can be $2$-tower group of real (and even imaginary) quadratic fields.

Theorem 5.7**.**

(Two-stage tower.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

Then the $2$-class tower of $K$ has $\ell_{2}(K)=2$ and the metabelian Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ is isomorphic to $\langle 512,1465\rangle-\#1;2$ of order $1024$.

Example 5.7**.**

$595\,561$ is the single known discriminant $0<d_{K}<10^{6}$ of a real quadratic field $K$ with $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ and Artin pattern in Formula (5.15). It is mentioned in [3, Exm. 7.4, p. 1192] without explicit invariant $\tau_{4}=(222)$.

Proof.

(Proof of Theorem 5.7) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$ and $\tau_{1}=(211,211,311)$, up to the first layer alone, yields $2$ groups of order $64$, $13$ groups of order $128$, for instance $\langle 128,25\rangle$, $21$ groups of order $256$, for instance $\langle 256,237\rangle$, and $56$ groups of order $512$, for instance $\langle 512,1465\rangle$. But none of the groups in the SmallGroups library hits the complete pattern up to the second layer, including $\tau_{2}=(31,31,31,31,311,422;2111)$, because $\tau_{2}(6)$ takes inadequate values in $\{311,321,322\}$, which are too small but compatible with $(422)$, by the monotony principle.

We can select the group $\langle 128,25\rangle$, resp. its child $\langle 256,237\rangle$, resp. its grand child $\langle 512,1465\rangle$, as the starting point for selectively constructing an extended pool of $2$-groups with orders bigger than $512$, and thus outside of the SmallGroups database. We generate descendants with depth three (including great grand children), resp. depth two (including grand children), resp. depth one (only immediate children), and step size $1$ alone, since there are no bifurcations to coclass bigger than $4$. Among the $54$, resp. $47$, resp. $4$, groups generated in this manner, there are always exactly $2$ hits of the complete pattern $\tau_{0}=(22)$, $\tau_{1}=(211,211,311)$, $\tau_{2}=(31,31,31,31,311,422;2111)$ and $\tau_{4}=(222)$ of abelian type invariants, but without considering any capitulation patterns $\varkappa_{1}$ and $\varkappa_{2}$.

Table 14 shows the development of the Artin pattern along the root path of the four descendants $\langle 512,1465\rangle-\#1;i$ with $1\leq i\leq 4$.

The groups $\langle 512,1465\rangle-\#1;i$ with $3\leq i\leq 4$ can be eliminated, because $\tau_{2}(6)=(322)$ instead of $(422)$. Now the capitulation type $\varkappa_{1}=(J_{0},J_{0},K_{1})$ and $\varkappa_{2}=(H_{3},H_{3},H_{2},H_{2},H_{1},J_{11};H_{1})$ is taken into account, and we are unambiguously led to $\langle 512,1465\rangle-\#1;2$, since $\langle 512,1465\rangle-\#1;1$ is discouraged. ∎

5.4. Smallest three-stage tower of Hilbert $2$-class fields

To find a $2$-class field tower $\mathrm{F}_{2}^{\infty}(K)/K$ of length $\ell_{2}(K)=3$ with minimal degree over a quadratic field $K$ having $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$ we changed the technique of investigation. Instead of beginning with Artin patterns of fields $K$ we started with a search for non-metabelian $2$-groups $G$ with $G/G^{\prime}\simeq C_{4}\times C_{4}$ of minimal order in the SmallGroups database [4]. The result was a batch of $16$ groups $G$ of order $256=2^{8}$ with identifiers $\langle 256,i\rangle$, where $469\leq i\leq 484$. It turned out that all these groups share a common second derived quotient (metabelianization) $G/G^{\prime\prime}$, and that the latter coincides with their common parent $\pi(G)\simeq\langle 128,122\rangle$.

This situation was called a simple tree topology of type parent and child in [14, § 5, pp. 89–92]. The root path of the occurring $2$-groups with orders up to $256$ is given by (5.16). It is free from any bifurcations. All descendants are of step size $1$, and the coclass $3$ remains stable.

[TABLE]

Since $d_{2}(\pi(G))=5$, the tower length in the following theorem is strictly proven.

Theorem 5.8**.**

(Minimal three-stage tower.) Let $K$ be a real quadratic field with Artin pattern

[TABLE]

The $2$-class tower of $K$ has length $\ell_{2}(K)=3$ and the metabelianization $G/G^{\prime\prime}=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$ of the Galois group $G=\mathrm{Gal}(\mathrm{F}_{2}^{\infty}(K)/K)$ is isomorphic to $\langle 128,122\rangle$ with coclass $3$ and relation rank $5$. The group $G$ itself could be one of the twelve immediate descendants $\langle 256,i\rangle$ with relation rank $4$ and coclass $3$, where $470\leq i\leq 472$ or $474\leq i\leq 476$ or $478\leq i\leq 480$ or $482\leq i\leq 484$.

Example 5.8**.**

In reverse direction, we used the common Artin pattern $\mathrm{AP}(G)$ of all these $12$ non-metabelian groups $G$ and of their metabelianization $G/G^{\prime\prime}$ in Formula (5.17) for finding quadratic fields $K$ having this pattern $\mathrm{AP}(K)=\mathrm{AP}(G)$. Indeed, we discovered two suitable real quadratic fields with the following minimal discriminants $d_{K}$:

[TABLE]

Proof.

(Proof of Theorem 5.8) Searching the SmallGroups database [4] for the pattern $\tau_{0}=(22)$ and $\tau_{1}=(211,211,211)$, up to the first layer alone, yields $1$ group of order $32$, $5$ groups of order $64$, $12$ groups of order $128$, $52$ groups of order $256$, and $218$ groups of order $512$. The complete pattern up to the second layer, including $\tau_{2}=(21,21,21,21,211,2111;1111)$, is hit by a single group of order $128$, namely $\langle 128,122\rangle$, $16$ groups of order $256$, namely $\langle 256,i\rangle$ with $469\leq i\leq 484$, $24$ groups of order $512$, namely $\langle 512,j\rangle$ with $1967\leq j\leq 1990$, and $16$ groups of order $1024$. They all share the correct capitulation type $\varkappa_{1}=(J_{0},J_{0},J_{0})$ and $\varkappa_{2}=(G,G,G,G,G,H_{3};H_{1})$, and minimal possible coclass $3$. The group of order $128$ is metabelian with relation rank $5$, all the other groups have derived length $3$. Only $12$, resp. $23$, groups of orders $256$ and $1024$, resp. $512$, are terminal with mandatory relation rank $4$. ∎

6. Conclusion

We have shown impressively that it is sufficient to compute the logarithmic abelian type invariants of $2$-class groups $\mathrm{Cl}_{2}(N)$ and the capitulation kernels of unramified abelian extensions $N/K$ in the first layer ($\tau_{1}$ and $\varkappa_{1}$) and second layer ($\tau_{2}$ and $\varkappa_{2}$) over the quadratic field $K$, that is, with absolute degrees $4$ and $8$. Together with the foregiven base layer $\tau_{0}=(22)$, corresponding to the general assumption that $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$, this information was the input for the pattern recognition process in our group theoretic search and the preamble of all theorems.

We never had to determine data for the third layer ($\tau_{3}$ and $\varkappa_{3}$), let alone for the Hilbert $2$-class field in the fourth layer ($\tau_{4}$), with absolute degrees $16$ and $32$. (Observe that the capitulation $\varkappa_{4}$ is total, according to the principal ideal theorem by Artin and Furtwängler.)

We emphasize that, on the contrary, we obtained the logarithmic abelian type invariants $\tau_{4}$ of the $2$-class group $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ of the Hilbert $2$-class field of $K$ as the structure of the commutator subgroup $G^{\prime}$ of the sifted metabelian $2$-group $G=\mathrm{Gal}(\mathrm{F}_{2}^{2}(K)/K)$. This is a significant progress in the determination of deeper arithmetical information compared with the paper of Benjamin and Snyder [3], where Pari/GP was unable to compute $\tau_{4}=(222)$ in Example 7.4 on page 1192.

The length of the $2$-class field tower of $K$ turned out to be either $\ell_{2}(K)=3$ or unexpectedly also $\ell_{2}(K)=2$ (in Theorem 4.9 and Theorem 5.7), when the rank of the $2$-class group $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ is three. We only found metabelian towers with length $\ell_{2}(K)=2$, when the rank of the $2$-class group $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ is two, although $\ell_{2}(K)=3$ should be possible, according to Blackburn [3, p. 1164].

7. Outlook on unsolved problems

We do not want to pretend the impression that this paper has covered all aspects of the $2$-class field tower of quadratic fields $K$ with $2$-class group $\mathrm{Cl}_{2}(K)\simeq C_{4}\times C_{4}$. We restricted our investigations mainly to the cases where Corollary 3.2 [3, p. 1182] by Benjamin and Snyder gives a warranty for a $2$-class group $\mathrm{Cl}_{2}(\mathrm{F}_{2}^{1}(K))$ of rank two of the Hilbert $2$-class field $\mathrm{F}_{2}^{1}(K)$ of $K$. In these cases, we always found a two-stage tower. We also touched upon some towers with three or even more stages, making clear that these questions lead to the difficult problem of higher abelian quotient invariants. We intentionally avoided the study of Artin patterns with high complexity, as given in Table 15, because there it is probably impossible to determine merely the metabelianization $G/G^{\prime\prime}$ of the $2$-tower group $G$.

8. Acknowledgements

We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): project P 26008-N25. We are indebted to Professor M. F. Newman from the Australian National University in Canberra for drawing our attention to the possibility of a four-stage tower in Theorem 4.7 and the finitely presented $2$-groups in Theorem 4.8.