\(3\)-class field towers with \(2\) or \(3\) stages

1. Our intention

We use logarithmic abelian type invariants of second order to state necessary and sufficient criteria for finite towers of successive maximal unramified abelian 3-extensions, so-called Hilbert \(3\)-class field towers, over quadratic number fields to possess a length of precisely two respectively three stages. The underlying quadratic fields have an elementary bicyclic \(3\)-class group and one of four well-known simple \(3\)-principalization types in their four unramified cyclic cubic extensions. In two main theorems, we state the outstanding new criteria, and we identify the graph theoretic position of crucial Galois groups. By means of challenging computations, our theory is underpinned by numerical prototypes of the four simple \(3\)-principalization types and both tower lengths. For two complex \(3\)-principalization types, we can only state theorems with partial conditions.

2. Introduction

Let \(k=\mathbb{Q}(\sqrt{d})\) be a quadratic number field with fundamental discriminant \(d\), elementary bicyclic \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\), and one of six \(3\)-principalization types [1, 2], [3, p, 91], also known as capitulation types or transfer kernel types (TKT),

\[\label{eqn:SimpleTKTComplexTKT} \begin{aligned} & \text{E.6: }\varkappa(k)\sim(1122), \qquad & \text{G.16: }\varkappa(k)\sim(4231), \\ & \text{E.8: }\varkappa(k)\sim(1231), \qquad & \text{H.4: }\varkappa(k)\sim(2122), \\ & \text{E.9: }\varkappa(k)\sim(2231), \qquad & \text{two \textbf{complex} types.} \\ & \text{E.14: }\varkappa(k)\sim(3122), \\ & \text{four \textbf{simple} types,} \end{aligned} \tag{1}\]

For the simple types, our goal is a necessary and sufficient criterion for the coincidence of the \(\sigma\)-automorphism group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) of the maximal unramified pro-\(3\)-extension of \(k\) and the \(\sigma\)-Galois group \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) of the maximal metabelian unramified \(3\)-extension of \(k\), independently of the so-called state, which is given in terms of the logarithmic abelian type invariants (ATI) \(\alpha(k)=(\alpha_i)_{i=1}^4\) of the \(3\)-class groups \(\alpha_i:=\mathrm{ATI}(\mathrm{Cl}_3(E_i))\) of the four unramified cyclic cubic extensions \(E_i/k\), \(1\le i\le 4\). When \(S=M\), then the \(3\)-class field tower of \(k\) has exactly two stages, \(\ell_3(k)=2\). In the case of non-coincidence, \(S\ne M\), we have precisely three stages, \(\ell_3(k)=3\), for simple types, but we have three or more stages, \(\ell_3(k)\ge 3\), for complex types. Then we determine the path between \(M\) and \(S\) in the descendant tree of either the root \(Q=\mathrm{SmallGroup}(729,49)\), briefly \(\langle 729,49\rangle\), for the types E.6, E.14 and H.4, or the root \(U=\mathrm{SmallGroup}(729,54)\), briefly \(\langle 729,54\rangle\), for the types E.8, E.9 and G.16 [4] (the designations non-CF group \(Q\) and \(U\) are due to Ascione et al. [5, Tbl. 1, p. 265, Tbl. 2, p. 266, Tbl. 3, p. 268]). We briefly speak about the Q-tree \(\mathcal{T}(Q)\) and the U-tree \(\mathcal{T}(U)\).

The present article analyzes Hilbert \(3\)-class field towers with a length \(\ell_3(k)\in\lbrace 2,3\rbrace\) for simple types, presumably unbounded length \(\ell_3(k)\ge 2\) for complex types, and a periodic group \(M\) of fixed coclass \(\mathrm{cc}(M)=2\), called second maximal class in [5], and odd nilpotency class \(\mathrm{cl}(M)\ge 5\) for simple types, even nilpotency class \(\mathrm{cl}(M)\ge 6\) for complex types. The metabelianization \(M=S/S^{\prime\prime}\) of \(S\) is located on one of two coclass trees with roots \(\langle 243,6\rangle\), the parent of \(\langle 729,49\rangle\), respectively \(\langle 243,8\rangle\), the parent of \(\langle 729,54\rangle\), in the form of vertices with depth \(\mathrm{dp}(M)=1\) for simple types, and \(\mathrm{dp}(M)=2\) for complex types. The \(n\)-th state of \(M\) is characterized by the ATI \[\label{eqn:States} \alpha(k)= \begin{cases} \lbrack(n+3,n+2),1^3,(21)^2\rbrack \text{ for the types E.6, E.14 and H.4, and} \\ \lbrack(n+3,n+2),(21)^3\rbrack \text{ for the types E.8, E.9 and G.16}, \end{cases} \tag{2}\] with heterocyclic polarization \((n+3,n+2)\), or also by odd class \(\mathrm{cl}(M)=5+2n\) for simple types and even class \(\mathrm{cl}(M)=6+2n\) for complex types, for each integer \(n\ge 0\), called the state parameter. In the case of non-coincidence, \(S\ne M\), we emphasize the remarkable fact that for each of these two descendant trees, there exists a non-metabelian infinite path with strictly alternating step sizes \(s=2\) and \(s=1\) and vertices of the skeleton type [3, p. 91] of the mainlines of the coclass trees, that is, c.18: \(\varkappa(k)\sim(0122)\), respectively c.21: \(\varkappa(k)\sim(0231)\), [6], beginning at the bifurcation \(Q=\langle 729,49\rangle\), respectively \(U=\langle 729,54\rangle\), from which the Schur \(\sigma\)-groups \(S\) [7] of the ground state \(n=0\) and of all excited states \(n\ge 1\) branch off as terminal leaves with step size \(s=2\) and depth \(\mathrm{dp}(S)=1\) for simple types and odd \(\mathrm{dp}(S)\ge 3\) for complex types.

We begin with arithmetic foundations in §3, §4 and group theoretic techniques in §5. The main results and proofs are contained in §7 and §9, ostensively visualized by tree diagrams in §10, and underpinned by computer experiments in §11 and prototypes in §12.

3. Terminology and notation

Let \(K/\mathbb{Q}\) be an algebraic number field. Given a prime number \(p\in\mathbb{P}\), we denote the Sylow \(p\)-subgroup \(\mathrm{Syl}_p\mathrm{Cl}(K)\) of the ideal class group \(\mathrm{Cl}(K)\) of \(K\) by \(\mathrm{Cl}_p(K)\), and we call it the \(p\)-class group of \(K\).

The concept \(p\)-class field was coined by Hilbert in his ‘‘Zahlbericht’’ [8, Satz 94, p. 279], since Hilbert conjectured an isomorphism of the automorphism group \(\mathrm{Gal}(\mathrm{F}_p^1(K)/K)\) to the \(p\)-class group \(\mathrm{Cl}_p(K)\) of \(K\). His hypothesis turned out to be a special case of the Artin reciprocity law [9, Allgemeines Reziprozitätsgesetz, p. 361].

The construction of Hilbert \(p\)-class fields may be iterated and leads to the maximal unramified pro-\(p\)-extension \(T\) of \(K\) with a potentially infinite topological pro-\(p\) Galois group \(\mathrm{Gal}(T/K)\), endowed with the Krull topology.

Generally, if \(\ell_p(K)\ge q\), then the Galois group \(G=\mathrm{Gal}(\mathrm{F}_p^q(K)/K)\) of the \(q\)-th \(p\)-class field of \(K\) has soluble length (or derived length) \(\mathrm{sl}(G)=q\) (or \(\mathrm{dl}(G)=q\)). In particular:

In the present article, our focus will be on criteria for the distinction between metabelian two-stage towers with \(\mathrm{F}_p^2(K)=\mathrm{F}_p^\infty(K)\), \(M=S\), and \(\ell_p(K)=2\), and on the other hand non-metabelian towers with length \(\ell_p(K)\ge 3\), in the situation with the particular prime \(p=3\).

4. Artin pattern and invariants of second order

In the Introduction §2, we used transfer kernel types (TKT) \(\varkappa(k)\) and logarithmic abelian type invariants (ATI) \(\alpha(k)\) of quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) to describe the goals of this article. Now we give precise definitions of these concepts for a quadratic number field \(k\) with fundamental discriminant \(d(k)=d\) and elementary bicyclic \(3\)-class group \(\mathrm{Cl}_3(k)\simeq (\mathbb{Z}/3\mathbb{Z})^2\). Such a field has four unramified cyclic cubic extensions \(E_i/k\), \(1\le i\le 4\). A natural order of these four extensions \(E_i\), which are dihedral of degree six over \(\mathbb{Q}\) and share a common discriminant \(d(E_i)=d^3\), is given by increasing regulators \(R_1<\ldots<R_4\) of the non-Galois absolutely cubic subfields \(L_i<E_i\).

The logarithmic notation is very compact, for instance, \(21\hat{=}(9,3)\hat{=}(\mathbb{Z}/9\mathbb{Z})\times(\mathbb{Z}/3\mathbb{Z})\), \(1^3\hat{=}(3,3,3)\hat{=}\) \((\mathbb{Z}/3\mathbb{Z})^3\), and \((n+3,n+2)\hat{=}(3^{n+3},3^{n+2})\hat{=}(\mathbb{Z}/3^{n+3}\mathbb{Z})\times(\mathbb{Z}/3^{n+2}\mathbb{Z})\), for any state parameter \(n\ge 0\).

5. Decisions by means of group theoretic constraints

5.1. Galois action

The decision, whether a finite metabelian \(p\)-group \(M\) is admissible as the second \(p\)-class group \(M=\mathrm{Gal}(\mathrm{F}_p^2(K)/K)\) of a quadratic number field \(K=\mathbb{Q}(\sqrt{d})\), for some odd prime \(p\), or not, can be made with the aid of group theoretic requirements: it is necessary that there exists an involutory automorphism \(\sigma\in\mathrm{Aut}(M)\) of order \(\mathrm{ord}(\sigma)=2\) which acts as the inversion \(x\mapsto x^{-1}\) on the first and second cohomology group \(H^i(M,\mathbb{F}_p)\), for \(i\in\lbrace 1,2\rbrace\). The same conditions are required for any pro-\(p\)-group \(S\) to be admissible as the full \(p\)-class field tower group \(S=\mathrm{Gal}(\mathrm{F}_p^\infty(K)/K)\) [12, Lem. (4.1) and (4.2), p. 217].

5.2. Relation rank

Let \(p\) be a prime number and let \(G\) be a pro-\(p\)-group. The dimension of cohomology groups of \(G\) acting on the finite field \(\mathbb{F}_p\) as a \(G\)-module is crucial for applications in class field theory.

In 1964, Shafarevich [13] determined bounds for the relation rank of the automorphism group \(G=\mathrm{Gal}(\mathrm{F}_{p,\mathcal{S}}^\infty(K)/K)\) of the maximal pro-\(p\)-extension \(\mathrm{F}_{p,\mathcal{S}}^\infty(K)\) of an algebraic number field \(K\) which is unramified outside of an assigned finite set \(\mathcal{S}\) of (real archimedean or non-archimedean) places. In the special case of the maximal (everywhere) unramified pro-\(p\)-extension \(\mathrm{F}_{p}^\infty(K)\) with \(\mathcal{S}=\emptyset\), the Shafarevich theorem states the following sharp estimate.

6. Finite bounds or potential infinitude?

There arises a basic partially mathematical and partially philosophical question: Which of our results may be claimed for any value of the state parameter \(n\) ?

\(\bullet\) It is clear that our experimental information about concrete numerical realizations of our theoretical claims are very limited, although they are based on the most extensive database of relevant arithmetical invariants for quadratic fields \(k=\mathbb{Q}(\sqrt{d})\) available currently. It is due to Michael Raymond Bush in 2015, see subsection §11.1. The highest excited state for which we know a single numerical example \(d=705\,576\,037\) of complex type H.4 is \(n=3\) (ES3). The biggest example \(d=336\,698\,284\) of a simple type E.14 has only \(n=2\) (ES2). The characteristic polarizations \(\alpha_1(k)\) of the states are the heterocyclic \(3\)-groups \((27,9)\) for \(n=0\) (GS), \((81,27)\) for \(n=1\) (ES1), \((243,81)\) for \(n=2\) (ES2), and \((729,243)\) for \(n=3\) (ES3), whereas no real quadratic examples are known with \((2187,729)\) for \(n=4\) (ES4). So the bound \(n\le 3\) for the state would be sufficient to explain all available experimental data concerning prototypes and other paradigms in §12.

\(\bullet\) Our number theoretic statements are derived from group theoretic theorems by means of the Artin reciprocity law [9]. So the former are subject to the same bounds as the latter.

\(\bullet\) By top-down techniques we understand the construction of finite \(3\)-quotients \(\mathcal{L}/\mathcal{U}\) of infinite limit groups \(\mathcal{U}\triangleleft\mathcal{L}\), or also the parametrized pc-presentation (power-commutator presentation) of an infinite sequence \((M_n^i)_{n\ge 0}\) of finite \(3\)-groups. Due to internal limitations of the maximal word-length in Magma [14], the bound \(n\le 7\) cannot be surpassed [3, Rmk 3.1, p. 95].

\(\bullet\) By bottom-up techniques we understand the construction of descendants \(D=P-\#s;i\) (with step-size \(s\) and counter \(i\)) of finite \(3\)-groups \(P=\pi(D)\) (with parent-projection \(\pi\)) by means of the \(p\)–group generation algorithm (or extension algorithm) by Newman and O’Brien [15– 17]. Here, no limitations of storage capacity (RAM) and no bounds for data types arise in Magma [14]. The amount of required CPU time, however, even on fast processors, suggests reasonable bounds \(n\le 19\) for difficult cases or at most \(n\le 29\) for easy cases (simple types) [3, Rmk 3.2, p. 96].

\(\bullet\) The connection between the state parameter \(n\) and logarithmic order (lo), nilpotency class (cl) and coclass (cc) is given in Table 1, for simple and complex types separately.

\(\bullet\) In spite of the finite bounds which are necessary for computations with computer algebra systems like Magma [14], it is our philosophical conviction that knowledge of either an infinite limit group or a parametrized pc-presentation justifies statements involving potential infinitude.

Based on this opinion, we now state our main theorems in section §7 and we give proofs in §8.

7. Length of \(3\)-class field towers

7.1. Length for simple types

There is a remarkable difference between the simple types E.6: \(\varkappa(k)\sim(1122)\), E.8: \(\varkappa(k)\sim(1231)\), E.9: \(\varkappa(k)\sim(2231)\), E.14: \(\varkappa(k)\sim(3122)\), in section E of the paper [18, p. 36] by Scholz and Taussky, and the complex types G.16: \(\varkappa(k)\sim(4231)\), H.4: \(\varkappa(k)\sim(2122)\) [18, pp. 36-38]. For simple types, a non-metabelian \(3\)-class field tower has precisely three stages, \(\ell_3(k)=3\), whereas for complex types, the length \(\ell_3(k)\ge 3\) is unbounded.

Theorem 2 only gives the connection between the length \(\ell_3(k)\) and the ATI2 \(\alpha^{(2)}(k)\). So we need additional information on the Galois groups \(M=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) and \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) in the case of non-coincidence, i.e., \(\ell_3(k)=3\). After some definitions, this will be done in Theorem 3.

A transfer kernel type (TKT) \(\varkappa=(\varkappa_1,\ldots,\varkappa_4)\) is called skeleton type if a single component \(\varkappa_i=0\) is a total capitulation, and hull type if all components \(\varkappa_i\in\lbrace 1,\ldots,4\rbrace\) are partial capitulations. For an imaginary quadratic field \(k\), the capitulation type \(\varkappa(k)\) must be a hull type.

In the sequel, we are going to use the ANUPQ-notation of iterated descendant groups by relative identifiers, ‘‘descendant \(=\) parent\(-\#s;j\)’’ [19] with step size \(1\le s\le\nu\), bounded by the nuclear rank \(\nu\), and counter \(1\le j\le N_s\), bounded by the number of \(s\)-descendants \(N_s\).

Theorem 3(Structure of 3-class field towers with precisely three stages for simple types).There exists a unique \(\sigma\)-group \(F\), the fork, with nilpotency class \(\mathrm{cl}(F)=4\) and nuclear rank \(\nu(F)=2\), causing a bifurcation, such that the path between the metabelian group \(M=S/S^{\prime\prime}\) with relation rank \(d_2(M)=3\) and the non-metabelian Schur \(\sigma\)-group \(S\) of relation rank \(d_2(M)=2\) in the descendant tree of \(\mathrm{SmallGroup}(9,2)=(\mathbb{Z}/3\mathbb{Z})^2\) is given by the union of the two paths (Figure 1).

\[\label{eqn:SimplePath} M=M_n^i:=F(-\#1;1-\#1;1)^n\lbrack-\#1;i\rbrack \quad \text{ and } \quad S=S_n^i:=F(-\#2;1-\#1;1)^n\lbrack-\#2;i\rbrack, \tag{12}\] where the parentheses \((\,)\) include vertices of skeleton type and the brackets \(\lbrack\,\rbrack\) include vertices of simple hull type. The final counter \(i=2,3,4\) depends on the capitulation type, E.6 for \(i=2\), E.14 for \(i=3,4\) on the \(Q\)-tree, and E.8 for \(i=2\), E.9 for \(i=3,4\) on the \(U\)-tree. The exponent \(n\) of the skeleton part expresses iteration and is the non-negative integer in the nilpotency class \(\mathrm{cl}(M)=\mathrm{cl}(S)=5+2n\), i.e., the state parameter. Finally, \(F\) is the fork between \(M\) and \(S\). In the case of a real quadratic field \(k\), the metabelian group \(M\) itself and, if \(n\ge 1\) (ES!), additional non-metabelian Schur\(+1\) \(\sigma\)-groups (all of them with relation rank \(d_2=3\)), are admissible for \(S\):

\[\label{eqn:SimplePathReal} S=T_{n,u}^i:=F(-\#2;1-\#1;1)^u(-\#1;1-\#1;1)^{n-u}\lbrack-\#1;i\rbrack \text{ with } 1\le u\le n, \text{ i.e.} \tag{13}\]

Ostensively: if \(v:=n-u\), then the pairs \((u,v)\) satisfy \(u+v=n\) and run through \((1,n-1),\ldots,(n,0)\).

The cover of the second \(3\)-class group \(M_n^i\) with \(i=2,3,4\) becomes bigger when the state parameter \(n\ge 0\) increases. It has \(n+1\) elements, and so it is a singleton in the ground state (GS), \(n=0\):

\[\label{eqn:CovMSimple} \mathrm{cov}(M_n^i)=\lbrace T_{n,u}^i\mid 1\le u\le n\rbrace\cup\lbrace S_n^i\rbrace \quad \text{ and } \quad \mathrm{cov}_\ast(M_n^i)=\lbrace S_n^i\rbrace. \tag{14}\]

7.2. Length for complex types

As mentioned at the beginning of the previous subsection §7.1, a non-metabelian \(3\)-class field tower has precisely three stages, \(\ell_3(k)=3\), for simple types, whereas the length \(\ell_3(k)\ge 3\) is unbounded for the complex types G.16: \(\varkappa(k)\sim(4231)\), H.4: \(\varkappa(k)\sim(2122)\) [18, pp. 36-38].

Theorem 5. (Structure of \(3\)-class field towers with three or more stages for complex types.) There exists a unique \(\sigma\)-group \(F\), the fork, with nilpotency class \(\mathrm{cl}(B)=4\) and nuclear rank \(\nu(B)=2\), causing bifurcation, such that the path between the metabelian group \(M=S/S^{\prime\prime}\) with relation rank \(d_2(M)=3\) and one of the infinitely many possible non-metabelian Schur \(\sigma\)-groups \(S\) of relation rank \(d_2(M)=2\) in the descendant tree of \(\mathrm{SmallGroup}(9,2)=(\mathbb{Z}/3\mathbb{Z})^2\) is given by the union of the two paths (Figure 2). \[\label{eqn:ComplexPath} \begin{aligned} M=M_n^i &= F(-\#1;1-\#1;1)^n\lbrack-\#1;i-\#1;1\rbrack \quad \text{ and } \quad \\ S=S_{n,t}^i &= F(-\#2;1-\#1;1)^n\lbrack-\#2;i-\#1;1\rbrack\lbrack-\#2;1-\#1;1\rbrack^{t}\lbrack-\#2;2\rbrack \end{aligned} \tag{19}\] where the parentheses \((\,)\) include vertices of skeleton type and the brackets \(\lbrack\,\rbrack\) include vertices of complex hull type. The intermediate counter \(i=5,6\) represents two possible variants, which cannot be distinguished arithmetically. The complex hull type is H.4 on the \(Q\)-tree and G.16 on the \(U\)-tree. The exponent \(n\) of the skeleton part is the non-negative integer in the nilpotency class \(\mathrm{cl}(M)=6+2n\), i.e., the state parameter, and the exponent \(t\) of the complex hull part is the non-negative integer in the nilpotency class \(\mathrm{cl}(S)=7+2t+2n\), i.e., the sub-state parameter. Finally, \(F\) is the fork between \(M\) and \(S\). In the case of a real quadratic field \(k\), additional Schur\(+1\) \(\sigma\)-groups are admissible for \(S\): Firstly, as immediate descendants of \(M\) with child topology,

\[\label{eqn:ComplexPathChild} S=R_{n,j}^i:=M_n^i-\#1;j \quad \text{ with } j=1,2,3 \text{ for type H.4 }, j=1,2 \text{ for type G.16}. \tag{20}\]

Secondly, via the fork \(F\) with fork topology, if \(n\ge 1\),

\[\label{eqn:ComplexPathFork} S=T_{n,u}^i:=F(-\#2;1-\#1;1)^u(-\#1;1-\#1;1)^{n-u}\lbrack-\#1;i-\#1;1\rbrack \text{ with } 1\le u\le n, \text{ i.e.} \tag{21}\]

Ostensively: if \(v:=n-u\), then the pairs \((u,v)\) satisfy \(u+v=n\) and run through \((1,n-1),\ldots,(n,0)\).

The cover of the second \(3\)-class group \(M_n^i\) with \(i=5,6\) is always infinite, and its components becomes bigger when the state parameter \(n\ge 0\) increases: \[\begin{aligned} \label{eqn:CovMComplex} \mathrm{cov}(M_n^i) = & \lbrace R_{n,j}^i\mid j=1,2(,3)\rbrace\quad\cup \lbrace\mathcal{T}_0(T_{n,u}^i)\mid 1\le u\le n\rbrace\quad\cup\lbrace\mathcal{T}_0(\pi(S_{n,t}^i))\setminus\mathcal{T}_0(\pi(S_{n,t+1}^i))\mid t\ge 0\rbrace \quad \text{ and } \notag\\ \mathrm{cov}_\ast(M_n^i) = & \lbrace S_{n,t}^i\mid t\ge 0\rbrace, \end{aligned} \tag{22}\] where \(\pi\) denotes the parent-operator, and \(\mathcal{T}_0\) is the pruned descendant tree restricted to \(\sigma\)-groups. The \(\mathcal{T}_0(T_{n,u}^i)\) are \(n\) finite saplings whose depth \(\mathrm{dp}=3+2u\) increases with \(u\). The differences \(\mathcal{T}_0(\pi(S_{n,t}^i))\setminus\mathcal{T}_0(\pi(S_{n,t+1}^i))\) are successive finite sections of the infinite branch \(\mathcal{T}_0(\pi(S_{n,0}^i))\), whose depth \(\mathrm{dp}=3+2n\) increases with \(n\). Their vertices conjecturally possess unbounded soluble length. The claims are proved for bounded state, parametrized by the integer \(n\) (ground state \(n=0\), excited states \(1\le n\le 4\)), but we conjecture they hold for any state \(n\ge 0\).

To reduce the complexity of the tree diagram, Schur\(+1\) \(\sigma\)-groups are not drawn in Figure 2. More details are drawn in Figure 3 for the ground state (GS) \(n=0\): the vertices \(R_{0,j}^i\) with child topology \(j=1,2,3\) for type H.4 (drawn), \(j=1,2\) for type G.16, and the finite section \(\mathcal{T}_0(\pi(S_{00}^i))\setminus\mathcal{T}_0(\pi(S_{01}^i))\) with depth \(\mathrm{dp}=3\) of the infinite branch \(\mathcal{T}_0(\pi(S_{00}^i))\) with periodic bifurcations. The vertices \(U_{0j}^i\), \(j=1,\ldots,4\), and \(V_{0}^i\) refer to both complex hull types, for the terminal leaves \(W_{0j}^i\), however, we have \(j=1,2,3\) for type H.4 (drawn), but \(j=1,2\) only for type G.16.

8. Proofs concerning the length of \(3\)-class field towers

First we show in two Propositions 1 and 2 how the main lines and branches down to depth \(\mathrm{dp}=1\) of both relevant trees, the \(Q\)-tree and the \(U\)-tree, can be given either by parametrized polycyclic pc-presentations or as quotients of infinite limit groups. These propositions concern the metabelianizations \(M\) with coclass \(\mathrm{cc}(M)=2\) of simple and complex types simultaneously.

The reason why only the \(Q\)-tree and the \(U\)-tree are relevant for our assigned six TKTs in Formula (1) is given with the aid of the ‘‘main theorem on class and coclass from IPAD’’ and its corollary [20, §4, Thm 2. and Cor. 1] in § 11.1 : after elimination of the finite \(3\)-groups \(G\) of maximal class, i.e., coclass \(\mathrm{cc}(G)=1\), among all descendants of the elementary bicyclic \(3\)-group \(\mathrm{SmallGroup}(9,2)=(\mathbb{Z}/3\mathbb{Z})^2\), we arrive at the seven groups in Hall’s isoclinism family \(\Phi_6\) by bifurcation from the extra-special \(3\)-group \(\langle 27,3\rangle\). Then the \(\varepsilon\)-invariant of the IPAD and the required shape of the six TKTs eliminates five of these groups, \(\langle 243,i\rangle\) with \(i=3,4,5,7,9\), and only \(\langle 243,6\rangle\), the parent of \(Q=\langle 729,49\rangle\), and \(\langle 243,8\rangle\), the parent of \(U=\langle 729,54\rangle\), remain admissible.

The following result shows that certain \(3\)-groups of class at least \(5\) on the coclass tree \(\mathcal{T}^2(\langle 243,6\rangle)\) belong to \(6+4=10\) periodic coclass sequences (6 odd, 4 even) with period length \(2\).

The following result shows that certain \(3\)-groups of class at least \(5\) on the coclass tree \(\mathcal{T}^2(\langle 243,8\rangle)\) belong to \(6+4=10\) periodic coclass sequences (6 odd, 4 even) with period length \(2\).

For the special case of the \(Q\)-tree and the \(U\)-tree, the two Propositions 1 and 2 provide an explicit proof of the periodicity of tree branches, proven generally and independently in [21– 25]. The finitely many pre-periodic instances down to class \(c=3,4\) have similar pc-presentations with slight degenerations, due to the low values of \(c\). However, our metabelianizations \(M\) in Theorem 3 and 5 have odd nilpotency class \(\mathrm{cl}(M)=2n+5\ge 5\), for each state \(n\ge 0\).

The complete theory of the mainlines with vertices of skeleton type, c.18, \(\varkappa\sim (0122)\), on the \(Q\)-tree, respectively c.21, \(\varkappa\sim (0231)\), on the \(U\)-tree, of the descendant tree \(\mathcal{T}(R)\) with root \(R\), containing an infinite main trunk of alternating step sizes \(s=2\) and \(s=1\), is based on the following infinite limit groups, due to Mike F. Newman.

The next theorem is a second crucial ingredient of this proof section, establishing the finiteness and structure of the cover for each metabelian \(3\)-group \(M\) with transfer kernel type in section \(\mathrm{E}\).

A pure bottom up approach without top down constructions, instead of using Algorithm 3.2 in [3, pp. 97-98], is able to reach coclass \(r=32\), nilpotency class \(c=63\), and logarithmic order \(r+c=95\), without surpassing internal limits of Magma, and thus provides even stronger support for Theorem 7, from the experimental point of view.

Currently, a drawback of the quotients \(\mathcal{Q}^{(0,+1)}_c\) with \(e=0\), \(\ell=+1\), \(c\ge 5\), which lead into a completely different realm, namely the complicated brushwood of the complex transfer kernel type \(\mathrm{H}.4\), is that they cannot be interpretated as covers of metabelianizations \(M_n^i\).

A common feature of all Theorems 2–5 together are certain general requirements for Galois groups \(\mathrm{Gal}(\mathrm{F}_p^q(k)/k)\), with a positive integer \(q\), of the stages of the \(p\)-class field tower with an odd prime \(p\), of quadratic number fields \(k=\mathbb{Q}(\sqrt{d})\).

Generally, any Galois group \(G=\mathrm{Gal}(\mathrm{F}_3^q(k)/k)\), with a positive integer \(q\), must be a \(\sigma\)-group, possessing an automorphism which acts as inversion on both cohomology groups \(H^i(G,\mathbb{F}_3)\), with \(i=1,2\), and the \(3\)-class field tower group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) additionally must satisfy the Shafarevich bound for the relation rank, \(2\le d_2(S)\le d_{2,\mathrm{max}}\) with \(d_{2,\mathrm{max}}=2\) for imaginary \(k\), enforcing a Schur \(\sigma\)-group, and \(d_{2,\mathrm{max}}=3\) for real \(k\), admitting also a Schur\(+1\) \(\sigma\)-group [6, 13].

For each simple type in section E [18, p. 36], and separately for each complex type in the sections G and H [18, pp. 36-38], the candidates for the metabelianization \(M=S/S^{\prime\prime}=\mathrm{Gal}(\mathrm{F}_3^2(k)/k)\) and for the \(3\)-class field tower group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) itself are identified by a search along the descendant tree of either the root \(Q=\langle 729,49\rangle\) or \(U=\langle 729,54\rangle\) [4, 5], constructed with the \(p\)-group generation algorithm by Newman [16, 26] and O’Brien [15, 17]. The claims for the ground state \(n=0\) and the lower excited states \(n=1,2,3,4\) are covered by initial computations with Magma [14, 27– 29], and evidence of statements for higher excited states \(n\ge 5\) is provided for the metabelianizations \(M_n^i\) either by Corollary 2 or by the periodicity of tree branches, which was proven independently in [21– 25].

In the notation with relative identifiers [19], and starting from one of the forks with bifurcation \(F\in\lbrace Q,U\rbrace\) as roots, for simple types, the metabelianizations turn out to be of the shape \(M=F(-\#1;1-\#1;1)^n\lbrack-\#1;i\rbrack\), and \(M=F(-\#1;1-\#1;1)^n\lbrack-\#1;i-\#1;1\rbrack\) for complex types, since, in the latter situation, for \(G=\pi(M)=F(-\#1;1-\#1;1)^n\lbrack-\#1;i\rbrack\), an automorphism acts as inversion on \(H^1(G,\mathbb{F}_3)\) only but not on \(H^2(G,\mathbb{F}_3)\). So the group \(M\) is still a vertex of the coclass tree \(\mathcal{T}^2(F)\) with exclusive step size \(s=1\) and thus with fixed coclass \(2\). It has depth \(\mathrm{dp}=1\) for simple types and \(\mathrm{dp}=2\) for complex types.

However, periodic bifurcations must actually be exploited for \(S=F(-\#2;1-\#1;1)^n\lbrack-\#2;i\rbrack\), if the type is simple, and for \(S=F(-\#2;1-\#1;1)^n\lbrack-\#2;i-\#1;1\rbrack\lbrack-\#2;1-\#1;1\rbrack^{t}\lbrack-\#2;2\rbrack\), when the type is complex, due to infinite branches with infinitely many candidates for each state \(n\ge 0\), expressed by the second parameter \(t\ge 0\).

8.1. Proofs for simple types

8.2. Proofs for complex types

9. Recall of simple types in the imaginary quadratic scenario, \(d<0\)

Suppose \(p\) is an odd prime number. Let \(k=\mathbb{Q}(\sqrt{d})\), with fundamental discriminant \(d<0\), be an imaginary quadratic number field possessing an elementary bicyclic \(p\)-class group \(\mathrm{Cl}_p(k)=(\mathbb{Z}/p\mathbb{Z})^2\). Then the Galois group \(S=\mathrm{Gal}(\mathrm{F}_p^\infty(k)/k)\) of the maximal unramified pro-\(p\)-extension of \(k\) is a non-abelian Schur \(\sigma\)-group [7] with balanced presentation and a generator and relator inverting automorphism \(\sigma\) of order \(2\) [30]. The second derived quotient \(S/S^{\prime\prime}\) of \(S\) is isomorphic to the Galois group \(M=\mathrm{Gal}(\mathrm{F}_p^2(k)/k)\) of the maximal metabelian unramified \(p\)-extension of \(k\).

For \(p=3\), we recall the well-known facts concerning the ground state (GS) and the excited states (ES) of \(3\)-tower groups \(S\) of \(k\) with capitulation type \(\varkappa(k)\) in section E of Scholz and Taussky [18]:

10. Tree diagrams

In both tree diagrams of Figure 1 and 2, there are two possible realizations of the root \(R\) on the top of the tree diagram. Both of these finite \(3\)-groups have nilpotency class \(\mathrm{cl}(R)=3\) and coclass \(\mathrm{cc}(R)=2\). In both cases, \(R\) belongs to Hall’s isoclinism family \(\Phi_6\) [33, p. 139]. On The \(Q\)-tree, the root is \(R=\langle 243,6\rangle\) with skeleton type c.18, \(\varkappa(k)\sim(0122)\). On The \(U\)-tree, the root is \(R=\langle 243,8\rangle\) with skeleton type c.21, \(\varkappa(k)\sim(0231)\).

10.1. The subtree \({}^\ast\mathcal{T}(R)\) pruned from complex types

Remarks concerning Figure 1. The figure is restricted to the root region up to logarithmic order \(\mathrm{lo}=20\), and to the pruned subtree \({}^\ast\mathcal{T}(R)<\mathcal{T}(R)\) of the complete descendant tree of the root \(R\), where the complex types are eliminated, and only simple types in section E of Scholz and Taussky [18] remain as terminal leaves, connected by mainline vertices of skeleton type. By \({}^\ast\mathcal{T}^2(R)\), we denote the pruned subtree of \({}^\ast\mathcal{T}(R)\) with fixed coclass \(\mathrm{cc}=2\), a so-called coclass tree. In terms of relative ANUPQ-identifiers [19], the Schur \(\sigma\)-groups [34] with simple types are given by periodic bifurcations as in Formula (12), \[S=S_n^i:=R(-\#1;1)(-\#2;1-\#1;1)^n\lbrack -\#2;i\rbrack,\ 2\le i\le 4,\] a unique Schur \(\sigma\)-group for each state \(n\ge 0\). Parentheses enclose vertices with skeleton type, and brackets surround vertices with simple types. The situation is unproblematic with respect to ambiguities, since the Schur \(\sigma\)-group \(S\) with \(M=S/S^{\prime\prime}\) is unique, but Schur\(+1\) \(\sigma\)-groups [35] with simple types also exist. The metabelianizations are given as second derived quotients as in Formula (12), \[M=M_n^i:=S/S^{\prime\prime}=R(-\#1;1)(-\#1;1-\#1;1)^n\lbrack -\#1;i\rbrack,\ 2\le i\le 4,\] on the pruned coclass subtree \({}^\ast\mathcal{T}^2(R)\).

The differences between the two possible realizations can be summarized as follows:

\(\bullet\) The \(Q\)-tree: It is pruned from the complex type H.4, \(\varkappa(k)\sim(2122)\), and the skeleton type of the mainline is c.18, \(\varkappa(k)\sim(0122)\). In particular, the root \(R=\langle 243,6\rangle\) and its immediate descendant, the fork \(Q=\langle 729,49\rangle=\langle 243,6\rangle(-\#1;1)\), are of skeleton type. The fork \(F=Q\) has a bifurcation, due to nuclear rank \(\nu=2\). The type of the terminal leaves is E.6, \(\varkappa(k)\sim(1122)\), for \(i=2\), and E.14, \(\varkappa(k)\sim(3122)\), for \(i=3,4\).

\(\bullet\) The \(U\)-tree: It is pruned from the complex type G.16, \(\varkappa(k)\sim(4231)\), and the skeleton type of the mainline is c.21, \(\varkappa(k)\sim(0231)\). In particular, the root \(R=\langle 243,8\rangle\) and its immediate descendant, the fork \(U=\langle 729,54\rangle=\langle 243,8\rangle(-\#1;1)\), possess the skeleton type. The fork \(F=U\) has a bifurcation to different step sizes \(s=1\) and \(s=2\). The type of the terminal leaves is E.8, \(\varkappa(k)\sim(1231)\), for \(i=2\), and E.9, \(\varkappa(k)\sim(2231)\), for \(i=3,4\).

10.2. The subtree \({}_\ast\mathcal{T}(R)\) pruned from simple types

Remarks concerning Figure 2. The figure is restricted to the root region up to logarithmic order \(\mathrm{lo}=20\), and to the pruned subtree \({}_\ast\mathcal{T}(R)<\mathcal{T}(R)\) of the complete descendant tree of the root \(R\), where the simple types in section E of Scholz and Taussky [18] are eliminated, and only complex types remain as finite or infinite branches, connected by mainline vertices of skeleton type. By \({}_\ast\mathcal{T}^2(R)\), we denote the pruned subtree of \({}_\ast\mathcal{T}(R)\) with fixed coclass \(\mathrm{cc}=2\), a so-called coclass tree.

The differences between the two possible realizations can be summarized as follows:

\(\bullet\) The \(Q\)-tree: It is pruned from the simple types, E.6, \(\varkappa(k)\sim(1122)\), and E.14, \(\varkappa(k)\sim(3122)\). The skeleton type of the mainline is c.18, \(\varkappa(k)\sim(0122)\). In particular, the root \(R=\langle 243,6\rangle\) and its immediate descendant, the fork \(Q=\langle 729,49\rangle=\langle 243,6\rangle(-\#1;1)\), are of skeleton type. The fork \(F=Q\) has a bifurcation, due to nuclear rank \(\nu=2\). The type of the infinite branches is the complex hull type H.4, \(\varkappa(k)\sim(2122)\).

\(\bullet\) The \(U\)-tree: It is pruned from the simple types, E.8, \(\varkappa(k)\sim(1231)\), and E.9, \(\varkappa(k)\sim(2231)\). The skeleton type of the mainline is c.21, \(\varkappa(k)\sim(0231)\). In particular, the root \(R=\langle 243,8\rangle\) and its immediate descendant, the fork \(U=\langle 729,54\rangle=\langle 243,8\rangle(-\#1;1)\), possess the skeleton type. The fork \(F=U\) has a bifurcation to different step sizes \(s=1\) and \(s=2\). The type of the infinite branches is the complex hull type G.16, \(\varkappa(k)\sim(4231)\).

10.3. Details for the complex ground state

Remarks concerning Figure 3. A detailed tree diagram for the ground state (GS) of complex types is drawn in Figure 3. It is pruned from simple types, and restricted to skeleton types and complex types. Four different scenarios can be realized by the same diagram: the two trees with fork \(F\) equal to \(Q=\langle 729,49\rangle\) or \(U=\langle 729,54\rangle\), and each of them with two possible selections of the metabelianization \(M_0^i\), sharing the same complex type in the GS.

\(\bullet\) The \(Q\)-tree: Skeleton type is c.18, \((0122)\), which forms the infinite mainline of the tree \(\mathcal{T}^2(R)\) with fixed coclass \(2\), starting at the root \(R=\langle 243,6\rangle\), passing the fork \(F=Q=\langle 729,49\rangle\) with nuclear rank \(\nu=2\) and bifurcation, on the one hand with permanent step size \(s=1\) to \(X_5^2=\langle 2187,285\rangle\), \(X_6^2=\langle 6561,2024\rangle\), etc. [36, Figure 3, p. 151], and on the other hand with alternating step sizes \(s=2\) and \(s=1\) on an infinite path with complex type H.4, \((2122)\), and with periodic bifurcations, giving rise to unboundedly increasing coclass. We are interested in two siblings of \(\langle 2187,285\rangle\) rather than in the mainline. They lead to two arithmetically indistinguishable candidates for the metabelianization \(M=\mathrm{Gal}(\mathrm{F}_3^2(K)/K)\) of the \(3\)-class field tower group.

1. Either via the sibling \(\langle 2187,286\rangle\), which is forbidden as \(M\) due to the lack of a proper \(\sigma\)-automorphism, to \(M=M_0^5=\langle 6561,2030\rangle\).

2. Or via the sibling \(\langle 2187,287\rangle\) which is forbidden as \(M\) due to the lack of a proper \(\sigma\)-automorphism, to \(M=M_0^6=\langle 6561,2035\rangle\).

In both cases, several configurations of \(3\)-class field tower groups \(S\) are possible. For real quadratic base fields \(K\), three terminal immediate non-metabelian descendants \(R_{0j}^i=M_0^i-\#1;j\), \(j=1,2,3\), provide the option of a Schur\(+1\) \(\sigma\)-group \(S\) with child topology.

Similarly, we are interested in two siblings, \(F-\#2;5=\langle 6561,614\rangle\) and \(F-\#2;6=\langle 6561,615\rangle\), of \(F-\#2;1=\langle 6561,613\rangle\) rather than in the skeleton type itself.

\(\bullet\) The \(U\)-tree: Skeleton type is c.21, \((0231)\), which forms the infinite mainline of the tree \(\mathcal{T}^2(R)\) with fixed coclass \(2\), starting at the root \(R=\langle 243,8\rangle\), passing the fork \(F=Q=\langle 729,54\rangle\) with nuclear rank \(\nu=2\) and bifurcation, on the one hand with permanent step size \(s=1\) to \(X_5^2=\langle 2187,303\rangle\), \(X_6^2=\langle 6561,2050\rangle\), etc. [36, Figure 4, p. 152], and on the other hand with alternating step sizes \(s=2\) and \(s=1\) on an infinite path with complex type G.16, \((4231)\), and with periodic bifurcations, causing unboundedly increasing coclass. We are interested in two siblings of \(\langle 2187,303\rangle\) rather than in the mainline. They lead to two arithmetically indistinguishable candidates for the metabelianization \(M=\mathrm{Gal}(\mathrm{F}_3^2(K)/K)\) of the \(3\)-class field tower group.

1. Either via the sibling \(\langle 2187,301\rangle\), which is forbidden as \(M\) due to the lack of a proper \(\sigma\)-automorphism, to \(M=M_0^5=\langle 6561,2048\rangle\).

2. Or via the sibling \(\langle 2187,305\rangle\) which is forbidden as \(M\) due to the lack of a proper \(\sigma\)-automorphism, to \(M=M_0^6=\langle 6561,2058\rangle\).

In both cases, several configurations of \(3\)-class field tower groups \(S\) are possible. For real quadratic base fields \(K\), two terminal immediate non-metabelian descendants \(R_{0j}^i=M_0^i-\#1;j\), \(j=1,2\), provide the option of a Schur\(+1\) \(\sigma\)-group \(S\) with child topology.

Similarly, we are interested in two siblings, \(F-\#2;5=\langle 6561,619\rangle\) and \(F-\#2;6=\langle 6561,623\rangle\), of \(F-\#2;1=\langle 6561,621\rangle\) rather than in the skeleton type itself.

11. Computational techniques

11.1. The fundamental database of Bush

On 11 July 2015, M. R. Bush kindly shared an extensive database with us, in private communication. This information was the computational background for the article [35], and extended the range of our own tables in [11, 37] by a factor of \(100\). It was created in several months of CPU time on a cluster of parallel supercomputers, and so it would have been impossible for us to reconstruct this numerical data. The associated README_real.txt file explains the contents of the database:

‘‘The Magma file ipad_freq_real.m contains two lists called disclist and ipadlist. Each list contains \(185\) entries.

Each entry in ipadlist is the IPAD, for \(p = 3\), of a real quadratic field \(K\) with \(3\)-class group of rank \(2\) and discriminant \(d_K < 10^9\). The IPAD of such a field \(K\) is a \(5\)-tuple in which the first entry is the \(3\)-class group of \(K\) and the remaining four entries are the \(3\)-class groups of the four unramified cyclic cubic extensions of \(K\).

The \(i\)-th entry in disclist is the complete list of discriminants \(d_K\) of real quadratic fields with \(3\)-class group of rank \(2\), \(d_K < 10^9\) and whose IPAD appears as the \(i\)-th entry in ipadlist.

All class groups have been computed using Magma V2.19-5.

Note. Taken together, the discriminants appearing in disclist form a complete list of discriminants for real quadratic fields \(K\) with \(3\)-class group of rank \(2\) and \(d_K < 10^9\). There are \(481\,756\) such fields.

Using a Magma program script SiftRealIPADs.m, we retrieved the following statistics of IPADs in Table 4, for the upper bound \(d_K < 10^9\), restricted to the elementary bicyclic first IPAD component \([ 3, 3 ]\): ‘‘Rel’’ denotes the relative counter of the \(35\) sifted IPADs, as opposed to the absolute counter ‘‘Abs’’ among all \(185\) IPADs. ‘‘NumDisc’’ is the number of discriminants with given IPAD. ‘‘MinDisc’’ is the minimal discriminant with assigned IPAD. Since the first component \([ 3, 3 ]\) is fixed, only the last four components of the IPAD are given. Boldface IPADs are crucial. Number of all IPADs: \(185\), Number of sifted IPADs: \(35\), Total number of discriminants: \(415\,698\).

In order to identify the relevant IPADs of vertices on the Q-tree and U-tree, corresponding to the six transfer kernel types under investigation (Formula 1), we define: Let \(\varepsilon\) be the number of elementary tricyclic components \([ 3, 3, 3 ]\) of an IPAD. By the ‘‘main theorem on class and coclass from IPAD’’ and its corollary in [20], it is well known that

• an IPAD with (at least) three elementary bicyclic components \([ 3, 3 ]\) is due to a group of maximal class, \(\mathrm{cc}=1\) (Abs = 1,2,4,6,25,32,74,86),

• an IPAD with \(\varepsilon=3\) is due to the sporadic group \(\langle 243,4\rangle\) and its descendants (Abs = 10),

• an IPAD with \(\varepsilon=2\) is either due to the sporadic Schur \(\sigma\)-group \(\langle 243,7\rangle\) (Abs = 9) or to the group \(\langle 243,3\rangle\) and its descendants, in particular to all groups with \(\mathrm{cc}\ge 3\) (Abs = 19,21,38,48,66,70,76,116,129,151),

• among the remaining IPADs with \(\varepsilon=1\), Abs = 5 is due to the sporadic Schur \(\sigma\)-group \(\langle 243,5\rangle\), the polarization of Abs = 12,45,114 is homocyclic, and only Abs = 17,59,165,170 belong to vertices with heterocyclic polarization on the Q-tree,

• among the remaining IPADs with \(\varepsilon=0\), Abs = 18 is due to the sporadic group \(\langle 243,9\rangle\) and its descendants, the polarization of Abs = 11,43,115 is homocyclic, and only Abs = 16,55,130 belong to vertices with heterocyclic polarization on the U-tree.

11.2. The principalization type

By means of GetDiscsForFixedIPAD.m, a Magma program script, we retrieved the discriminants in ascending order for the IPADs \[\textbf{Abs = 16,17,55,59,130,165,170}.\]

The TKT was computed with RQGroundDisc.m, the ATI2 and Length with SecondATIExtended.m.

In Table 5, the initial 15 discriminants with IPAD Abs = 16 \(( [ 3, 9 ], [ 3, 9 ], [ 3, 9 ], [ 9, 27 ] )\) are classified according to their transfer kernel type TKT E.8 or E.9 or G.16, in the ground state GS, \(n = 0\). The corresponding metabelianizations \(M\) are vertices on the U-tree.

In Table 6, the initial 10 discriminants with IPAD Abs = 17 \(( [ 3, 3, 3 ], [ 3, 9 ], [ 3, 9 ], [ 9, 27 ] )\) are classified according to their transfer kernel type TKT E.6 or E.14 or H.4, in the ground state GS, \(n = 0\). The corresponding metabelianizations \(M\) are vertices on the Q-tree.

In Table 7, the initial 18 discriminants with IPAD Abs = 55 \(( [ 3, 9 ], [ 3, 9 ], [ 3, 9 ], [ 27, 81 ] )\) are classified according to their transfer kernel type TKT E.8 or E.9 or G.16, in the first excited state ES 1, \(n = 1\). The corresponding metabelianizations \(M\) are vertices on the U-tree.

In Table 8, the initial 7 discriminants with IPAD Abs = 59 \(( [ 3, 3, 3 ], [ 3, 9 ], [ 3, 9 ], [ 27, 81 ] )\) are classified according to their transfer kernel type TKT E.6 or E.14 or H.4, in the first excited state ES 1, \(n = 1\). The corresponding metabelianizations \(M\) are vertices on the Q-tree.

12. Real quadratic prototypes

For each assigned transfer kernel type \(\varkappa(k)\in\lbrace (1122), (2122), (3122), (1231), (2231), (4231)\rbrace\) and each foregiven excited state of (logarithmic) abelian type invariants (ATI), or transfer target type (TTT), \(\alpha(k)\in\lbrace \lbrack(n+3,n+2),111,21,21\rbrack; \lbrack(n+3,n+2),21,21,21\rbrace\) with non-negative integers \(n\ge 0\), the prototype is the minimal positive quadratic fundamental discriminant \(d\) such that the real quadratic field \(k=\mathbb{Q}(\sqrt{d})\) has TKT \(\varkappa(k)\) and TTT \(\alpha(k)\).

The Ground State is characterized by the polarization \((32)\) in the ATI of the first order \[\label{eqn:Ground} \alpha(k)=\lbrack 32,111,21,21\rbrack, \text{ respectively } \lbrack 32,21,21,21\rbrack, \text{ see Table } 9\text{ under the GRH}, \tag{37}\] whereas the stabilization \((111,21,21)\), respectively \((21,21,21)\), remains the same for all states.

The First Excited State is characterized by the polarization \((43)\) in the ATI of the first order

\[\label{eqn:Excited} \alpha(k)=\lbrack 43,111,21,21\rbrack \text{ respectively } \lbrack 43,21,21,21\rbrack, \text{ see Table }10\text{ under the GRH}. \tag{38}\]

The root region up to the logarithmic order \(\mathrm{lo}=20\) of two infinite descendant trees [16, 17, 26] is drawn in Figure 1 for the pruned tree \({}^\ast\mathcal{T}(R)\), which is restricted to the skeleton type and simple types, and in Figure 2 for the pruned tree \({}_\ast\mathcal{T}(R)\), which is restricted to the skeleton type and the complex type. For the types E.6, E.14 and H.4, the root is \(R=\langle 243,6\rangle\) with type c.18, and for the types E.8, E.9 and G.16, the root is \(R=\langle 243,8\rangle\) with type c.21.

13. Conclusion

The investigation of non-metabelian \(3\)-class field towers with length \(\ell_3(k)=3\) started in 2012 in cooperation with M. R. Bush [10, Thm. 4.1 and Cor. 4.1.1, pp. 774-775], where the erroneous claim \(\ell_3(k)=2\) for the imaginary quadratic field \(k=\mathbb{Q}(\sqrt{-9748})\) with TKT E.9 in the GS \(n=0\) in [18, p. 41] was disproved by the rigorous verification that either \(S=S_0^3\) or \(S=S_0^4\), rather than \(S=M\), presented at the West Coast Number Theory Conference (WCNT) in Asilomar, December 2013.

In 2015, [31, pp. 184-193] we extended the proof for \(\ell_3(k)=3\) of 2012 to all four simple TKTs, E.6, E.8, E.9, E.14, of imaginary quadratic fields, up to ES \(n\le 7\), i.e. \(\mathrm{cl}\le 19\), \(\mathrm{cc}\le 10\), \(\mathrm{lo}\le 29\), introducing the cover \(\mathrm{cov}(M)\), presented at the 29th Journées Arithmétiques (JA) in Debrecen, July 2015. We also drew a more detailed version of the present Figure 1 (bounded by \(\mathrm{lo}=20\)) in the Figures 8–9 on pp. 188–189 (bounded by \(\mathrm{lo}=14\)), and we studied the normal lattice of the Schur \(\sigma\)-groups \(S_0^i\) and \(S_1^i\) in the Figures 10–11 on pp. 191–192.

Arithmetical prototypes up to ES \(n\le 4\) for all imaginary quadratic fields \(k=\mathbb{Q}(\sqrt{d})\), \(-10^8<d<0\), were computed for the Figures 3–4 on pp. 754–755 in [32], presented at the 1st International Conference on Goups and Algebras (ICGA) in Shanghai, July 2015. See also the Figures 1–2 on pp. 24–25 in [6].

It should be pointed out that the present Theorem 2 on the length \(\ell_3(k)\) of real quadratic fields with simple TKT was proved on 5 Oct 2025 for all excited states (ES \(n\)) with \(n\ge 1\), whereas it was known for the ground state (GS) with \(n=0\) already in [39, Thm. 6.3, pp. 298–299] for the \(U\)-tree alone, in [36, Thm. 7.5–7.12, pp. 159–166] for both, the \(Q\)-tree and the \(U\)-tree, and the trees with minimal positive discriminants \(d>0\) of prototypes were drawn in [36, Figure 3–4, pp. 151–152].

Due to the highly confusing branches of complex TKTs, Theorem 4 was proved on 15 Oct 2025 for bounded state parameter \(0\le n\le 4\) only. With second order ATIs, only implications can be proven rather than equivalences. TODO: For an extension to \(n\ge 5\), modified infinite limit groups would be required, maybe \(3\)-adic Lie groups like \(\mathrm{SL}_2(\mathbb{Z}_3)\).

It was possible to prove necessary and sufficient criteria for all excited states (ES) parametrized with state parameter \(n\ge 0\). Figure 1 ostensively reveals that, in the case of non-coincidence, \(S\ne M\), the connecting path between \(M\) and \(S\) consists of an ascending part with exclusive step size \(s=1\) from the metabelianization \(M=S/S^{\prime\prime}\) to the fork \(F=R(-\#1;1)\), and a descending part with alternating step sizes \(s=2\) and \(s=1\), the so-called structure elements (SE), from the fork \(F\) to the Schur \(\sigma\)-group \(S\). The ground state (GS) gives rise to logarithmic order \(\mathrm{lo}(S)=8\) without SE, and the \(n\)-th excited state (ES \(n\)) requires exactly \(n\) SE and leads to logarithmic order \(\mathrm{lo}(S)=8+3n\).

Examples for the distinction between the two possible tower lengths \(\ell_3(k)\in\lbrace 2,3\rbrace\) for real quadratic base fields \(k\) are given for the ground state (GS) in [36, Thm. 7.8, pp. 162–163, and Thm. 7.12, pp. 165–166]. For the first excited state (ES \(1\)), we have for instance three stages for \(d=70\,539\,596\) of type E.14, and for \(d=75\,393\,861\) of type E.6, but only two stages for \(d=26\,889\,637\) of type E.8. For the second excited state (ES \(2\)), we have only two stages for \(d=336\,698\,284\) of type E.14 (ES 2 is not contained in our tables).

The tree structure in Figure 2 is more complicated, due to infinite branches. In contrast to the simple types in section E of Scholz and Taussky, where a real quadratic field \(k=\mathbb{Q}(\sqrt{d})\) with tower length \(\ell_3(k)=3\) has a Schur or Schur\(+1\) \(\sigma\)-group \(S=\mathrm{Gal}(\mathrm{F}_3^\infty(k)/k)\) as the Galois group of its maximal unramified pro-\(3\)-extension and the path between \(S\) and \(M=S/S^{\prime\prime}\) is a fork topology as described above, the complex types in the sections G and H of Scholz and Taussky admit a variety of different scenarios, most frequently a child topology with an immediate (Schur+1)-descendant \(S=M-\#1\) of \(M\) and thus necessarily precise length \(\ell_3(k)=3\), for instance \(d=2\,747\,001\) and \(d=126\,691\,957\) of type H.4, and \(d=8\,711\,453\) of type G.16, or less frequently a fork topology with either a (Schur+1) \(\sigma\)-group \(S\), such as \(d=23\,064\,965\) and \(d=216\,353\,320\) of type H.4, and \(d=209\,483\,033\) of type G.16, or quite sparsely with a Schur \(\sigma\)-group \(S\), e.g. \(d=30\,118\,269\) of type H.4. All the latter are located on infinite branches with presumably unbounded \(\ell_3(k)\ge 3\).

In [40], a sporadic variant of the TKT H.4, \(\varkappa=(4111)\sim (2122)\), with stable ATI \(\alpha=\lbrack (1^3)^3,21\rbrack\) is analyzed with the aid of the special linear group of dimension \(2\) over the \(3\)-adic integers, \(\mathrm{SL}_2(\mathbb{Z}_3)\). It is the \(3\)-principalization type of all members of the infinite descendant tree of the root \(\langle 243,4\rangle\), whose unique immediate \(\sigma\)-descendant is Ascione’s group \(N=\langle 729,45\rangle\) [5]. The pruned subtree \(\mathcal{T}_\sigma(N)\) of all \(\sigma\)-descendants of \(N\) contains an infinite sequence \((G_m)_{m\ge 2}\) of non-metabelian Schur \(\sigma\)-groups possessing an unbounded soluble length \(\mathrm{sl}(G_m)=\lfloor\log_2(3m+3)\rfloor\) [40, Thm. 2.1, p. 160].

The location of these groups in the tree is given by the parametrized relative identifier \[\label{eqn:BaBu} N(-\#2;1-\#1;1)^{m-2}-\#2;2, \qquad m\ge 2. \tag{39}\]

TODO: It seems plausible that other \(3\)-adic Lie groups can be used to investigate the Schur \(\sigma\)-groups \(S_{n,t}^i\) with the periodic variant of the TKT H.4, \(\varkappa=(2122)\), and with variable ATI depending on the order \(n\ge 0\) of the excited state (ES \(n\)): \(\alpha=\lbrack (n+3,n+2),1^3,(21)^2\rbrack\) with heterocyclic polarization \((n+3,n+2)\).

In Table 11, we compare the logarithmic order lo, nilpotency class cl, coclass cc, soluble length sl, and the \(3\)-valuation \(a:=v_3(A_m)\) of the order of the automorphism group \(A_m:=\mathrm{ord}(G_m)\) with the corresponding values of the ground state (GS) and the first three excited states (ES) of the Schur \(\sigma\)-groups \(S_{n,t}^i\). TODO: Deterministic laws for their soluble length seem plausible.

Acknowledgments

We are indebted to Michael Raymond Bush at the Washington and Lee University (WLU) in Lexington, Virginia, USA, for making available an extensive database of quadratic fundamental discriminants with assigned abelian type invariants of the first order. Sincere thanks are given to Mike F. Newman at the Australian National University (ANU) in Canberra, Capital Territory, Australia, for the infinite limit groups which are crucial in the proofs of the main theorems. We thank Bill Allombert at the University of Bordeaux, Aquitaine, France, for computational aid in determining decisive abelian type invariants of the second order with PARI/GP for Table 10. The authors are deeply indebted to the anonymous referee for his expertise and helpful suggestions concerning the entire exposition.

Funding Information

The second author gratefully acknowledges financial support by the Austrian Science Fund (FWF): projects J0497-PHY, P26008-N25, and by the Research Executive Agency of the European Union (EUREA): project Horizon Europe 2021–2027.

Conflicts of Interest

The authors declare that no conflict of interests arises from this article.

Data Availability

Magma program scripts and output may be requested from the second author via email.