Criteria for three-stage towers of p-class fields

Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K), that is the second derived quotient M=G/G". The family tau(1,K) of abelian type invariants of the p-class groups Cl(p,L) of all unramified cyclic extensions L/K of degree p is called the index-p abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD ((tau(1,K),tau(2,K)) includes data on unramified abelian extensions L/K of degree p^2 and enables sharper bounds for the order of M in the case Cl(p,K)=(p,p,p), where current implementations of the p-group generation algorithm fail to produce explicit contestants for M, due to memory limitations. The iterated IPAD of second order tau^(2)(K) contains information on non-abelian unramified extensions L/K of degree p^2, or even p^3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(squareroot(d)) with Cl(p,K)=(p,p) possessing a p-class field tower of exact length L(p,K)=3 as a striking novelty.


Introduction
In a previous article [1], we provided a systematic and rigorous introduction of the concepts of abelian type invariants and iterated IPADs of higher order. These ideas were communicated together with impressive numerical applications at the 29th Journées Arithmétiques in Debrecen, July 2015 [2]. The purpose and the organization of the present article, which considerably extends the computational and theoretical results in [1,2], is as follows.
Index-p abelianization data (IPADs) are explained in § 2. Our Main Theorem on three-stage towers of 3-class fields is communicated in § 3. Basic definitions concerning the Artin transfer pattern [1,3,4] are recalled in § 4. Then we generally put p = 3 and consider 3-class tower groups ( § 7). In § 5, we first restate a summary of all possible IPADs of a number field K with 3-class group Cl 3 K of type (3,3) [1, Thm. 3.1-3.2, pp. 290-291] in a more succinct and elegant form avoiding infinitely many exceptions, and emphasizing the role of two distinguished components, called the polarization and co-polarization, which are crucial for proving the finiteness of the batch of contestants for the second 3-class group M = G 2 3 K. Up to now, this is the unique situation where all IPADs can be given in a complete form, except for the simple case of a number field K with 2-class group Cl 2 K of type (2,2) [5, § 9, pp. 501-503]. We characterize all relevant finite 3-groups by IPADs of first and second order in § § 7.1, 7.3, 7.6, 7.9. These groups constitute the candidates for 3-class tower groups G ∞ 3 K of quadratic fields K = Q( √ d) with 3-class group Cl 3 K of type (3,3). In § 7.2, results for the dominant scenario with 3-principalization κ 1 K of type a are given. In § § 7.5, 7.8, we provide evidence of unexpected phenomena revealed by real quadratic fields K with types κ 1 K in Scholz and Taussky's section E [6, p. 36]. Their 3-class tower can be of length 2 ≤ ℓ 3 K ≤ 3 and a sharp decision is possible by means of iterated IPADs of second order. We point out that imaginary quadratic fields with type E must always have a tower of exact length ℓ 3 K = 3 [3,7]. In § § 7.10, 7.11, resp. § § 7.12, 7.13, results for quadratic fields K with 3-principalization type H.4, κ 1 K ∼ (4111), resp. G. 19, κ 1 K ∼ (2143), are proved.
In the last section § 8 on multi-layered IPADs, it is our endeavour to point out that the rate of growth of successive derived quotients G n p K ≃ G/G (n) , n ≥ 2, of the p-class tower group G = G ∞ p K is still far from being known for imaginary quadratic fields K with p-class rank ̺ ≥ 3, where the criterion of Koch and Venkov [8] ensures an infinite p-class tower with ℓ p K = ∞.

Index-p abelianization data
Let p be a prime number. According to the Artin reciprocity law of class field theory [9], the unramified cyclic extensions L/K of relative degree p of a number field K with non-trivial p-class group Cl p K are in a bijective correspondence to the subgroups of index p in Cl p K. Their number is given by p ̺ −1 p−1 if ̺ denotes the p-class rank of K [10, Thm. 3.1]. The reason for this fact is that the Galois group G 1 p K := Gal(F 1 p K/K) of the maximal unramified abelian p-extension F 1 p K/K, which is called the first Hilbert p-class field of K, is isomorphic to the p-class group Cl p K. The fields L are contained in F 1 p K and each group Gal(F 1 p K/L) is of index p in G 1 p K ≃ Cl p K. It was also Artin's idea [11] to leave the abelian setting of class field theory and to consider the second Hilbert p-class field F 2 p K = F 1 p (F 1 p K), that is the maximal unramified metabelian pextension of K, and its Galois group M := G 2 p K := Gal(F 2 p K/K), the so-called second p-class group of K [5], [6, p. 41], for proving the principal ideal theorem that Cl p K becomes trivial when it is extended to Cl p (F 1 p K) [12]. Since K ≤ L ≤ F 1 p K ≤ F 1 p L ≤ F 2 p K is a non-decreasing tower of normal extensions for any assigned unramified abelian p-extension L/K, the p-class group of L, Cl p L ≃ Gal(F 1 p L/L) ≃ Gal(F 2 p K/L) Gal(F 2 p K/F 1 p L), is isomorphic to the abelianization H/H ′ of the subgroup H := Gal(F 2 p K/L) of the second p-class group Gal(F 2 p K/K) which corresponds to L and whose commutator subgroup is given by H ′ = Gal(F 2 p K/F 1 p L). In particular, the structure of the p-class groups Cl p L of all unramified cyclic extensions L/K of relative degree p can be interpreted as the abelian type invariants of all abelianizations H/H ′ of subgroups H = Gal(F 2 p K/L) of index p in the second p-class group Gal(F 2 p K/K), which has been dubbed the index-p abelianization data, briefly IPAD, τ 1 K of K by Boston, Bush, and Hajir [13]. This kind of information would have been incomputable and thus useless about twenty years ago. However, with the availability of computational algebra systems like PARI/GP [14] and MAGMA [15,16,17] it became possible to compute the class groups Cl p L, collect their structures in the IPAD τ 1 K, reinterpret them as abelian quotient invariants of subgroups H of G 2 p K, and to use this information for characterizing a batch of finitely many p-groups, occasionally even a unique p-group, as contestants for the second p-class group M = G 2 p K of K, which in turn is a two-stage approximation of the (potentially infinite) pro-p group G := G ∞ p K := Gal(F ∞ p K/K) of the maximal unramified pro-p extension F ∞ p K of K, that is its Hilbert p-class tower. As we proved in the main theorem of [4,Thm. 5.4], the IPAD is usually unable to permit a decision about the length ℓ := ℓ p K of the p-class tower of K when non-metabelian candidates for G ∞ p K exist. For solving such problems, iterated IPADs τ (2) K of second order are required.

The p-principalization type
Until very recently, the length ℓ of the p-class tower K < F 1 p K < F 2 p K < . . . F ℓ p K = F ℓ+1 p K = . . . = F ∞ p K over a quadratic field K = Q( √ d) with p-class rank ̺ = 2, that is, with p-class group Cl p K of type (p u , p v ), u ≥ v ≥ 1, was an open problem. Apart from the proven impossibility of an abelian tower with ℓ = 1 [5,Thm. 4.1.(1)], it was unknown which values ℓ ≥ 2 can occur and whether ℓ = ∞ is possible or not. In contrast, it is known that ℓ = 1 for any number field K with p-class rank ̺ = 1, i.e., with non-trivial cyclic p-class group Cl p K, and that ℓ = ∞ for an imaginary quadratic field with p-class rank ̺ ≥ 3, when p is odd [8].
The finite batch of contestants for M = G 2 p K, specified by the IPAD τ 1 K, can be narrowed down further if the p-principalization type of K is known. That is the family κ 1 K of all kernels ker T K,L of p-class transfers T K,L : Cl p K → Cl p L from K to unramified cyclic superfields L of degree p over K. In view of the open problem for the length of the p-class tower, there arose the question whether each possible p-principalization type κ 1 K of a quadratic field K with Cl p K of type (p, p) is associated with a fixed value of the tower length ℓ p K.
Concerning the steps for the proof, we provide information in the form of Table 1. An asterisk indicates the present paper. The last step has been completed in collaboration with M. F. Newman but has not been published yet [20]. Only the types G.16 and G.19 must be distinguished by their integer identifier, otherwise the types denoted by the same letter behave completely similar. Additionally, we give the smallest logarithmic order lo(G) := log 3 |G|.
5. All possible IPADs of 3-groups of type (3,3) Since the abelian type invariants of certain IPAD components of an assigned 3-group G depend on the parity of the nilpotency class c or coclass r, a more economic notation, which avoids the tedious distinction of the cases odd or even, is provided by the following definition [24, § 3].
In the following theorem and in the whole remainder of the article, we use the identifiers of finite 3-groups up to order 3 8 as they are defined in the SmallGroups Library [25,26]. They are of the shape order, counter , where the counter is motivated by the way how the output of descendant computations is arranged in the p-group generation algorithm by Newman [27] and O'Brien [28]. (3,3) and metabelianization M = G/G ′′ of nilpotency class c = cl(M) ≥ 2, defect 0 ≤ k = k(M) ≤ 1, and coclass r = cc(M) ≥ 1. Assume that M does not belong to the finitely many exceptions in the list below. Then the IPAD τ (1) G = [τ 0 G; τ 1 G] of G in terms of nearly homocyclic abelian 3-groups is given by where the polarized first component of τ 1 G depends on the class c and defect k, the co-polarized second component increases with the coclass r, and the third and fourth component are completely stable for r ≥ 3 but depend on the coclass tree of M for 1 ≤ r ≤ 2 in the following manner Anomalies of finitely many, precisely 13, exceptional groups are summarized in the following list. Proof. Equations (5.1) and (5.2) are a succinct form of information which summarizes all statements about the first TTT layer τ 1 G in the formulas (19), (20) and (22)  The abelian group 9, 2 ≃ (3, 3), the extra special group 27, 4 , and the group 81, 7 ≃ Syl 3 (A 9 ) do not fit into the general rules for 3-groups of coclass 1. These three groups appear in the top region of the tree diagram in the Figures 1 and 2.
Remark 5.1. The reason why we exclude the second TTT layer τ 2 G from Theorem 5.1, while it is part of [1, Thm. 3.1-3.2, pp. 290-291], is that we want to reduce the exceptions of the general pattern to a finite list, whereas the irregular case of the abelian quotient invariants of the commutator subgroup G ′ , which forms the single component of τ 2 G, occurs for each even value of the coclass r = cc(M) ≡ 0 (mod 2) and thus infinitely often.  Tables and Figures of possible 3-groups G 2 3 K and G ∞ 3 K 6.1. Tables. In this article, we shall frequently deal with finite 3-groups G of huge orders |G| ≥ 3 9 for which no identifiers are available in the SmallGroups database [25,26]. For instance in Table  6, and in the Figures 6 and 7. A work-around for these cases is provided by the relative identifiers of the ANUPQ (Australian National University p-Quotient) package [31] which is implemented in our licence of the computational algebra system MAGMA [15,16,17] and in the open source system GAP [32]. Recall that a group with nuclear rank ν = 0 is a terminal leaf without any descendants.
All numerical results in this article have been computed by means of the computational algebra system MAGMA [15,16,17]. The p-group generation algorithm by Newman [27] and O'Brien [28] was used for the recursive construction of descendant trees T (R) of finite p-groups G. The tree root (starting group) R was taken to be 9, 2 for Table 2 Table 3 and Figure 3, 243, 8 for Table 4 and Figure 4, 243, 4 for Table 5 and Figure 6, and 243, 9 for Table 6 and Figure 7. For computing group theoretic invariants of each tree vertex G, we implemented the Artin transfers T G,H from a finite p-group G of type G/G ′ ≃ (p, p) to its maximal subgroups H G in a MAGMA program script as described in [ [33]. They are discussed thoroughly in the broadest detail in the initial sections of [30]. Trees are crucial for the recent theory of p-class field towers [34,35,36], in particular for describing the mutual location of G 2 3 K and G ∞ 3 K. Generally, the vertices of coclass trees in the Figures 1, 2, 3, 4, of the sporadic part of a coclass graph in Figure 5, and of the descendant trees in the Figures 6, 7 represent isomorphism classes of finite 3-groups. Two vertices are connected by a directed edge G → H if H is isomorphic to the last lower central quotient G/γ c (G), where c = cl(G) denotes the nilpotency class of G, and either |G| = 3|H|, that is, γ c (G) is cyclic of order 3, or |G| = 9|H|, that is, γ c (G) is bicyclic of type (3,3). See also [29, § 2.2, p. 410-411] and [30, § 4, p. 163-164].
The vertices of the tree diagrams in Figure 1  (1) big contour squares represent groups G with relation rank d 2 G) ≤ 3, (2) small contour squares represent groups G with relation rank d 2 (G) ≥ 4. A symbol n * adjacent to a vertex denotes the multiplicity of a batch of n siblings, that is, immediate descendants sharing a common parent. The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library [25,26] of GAP [32] and MAGMA [17]. We omit the orders, which are given on the left hand scale. The IPOD κ 1 [18, Thm. 2.5, Tbl. [6][7], in the bottom rectangle concerns all vertices located vertically above. The first, resp. second, component τ 1 (1), resp. τ 1 (2), of the IPAD [1, Dfn. 3.3, p. 288] in the left rectangle concerns vertices G on the same horizontal level with defect k(G) = 0. The periodicity with length 2 of branches, B(j) ≃ B(j + 2) for j ≥ 4, resp. j ≥ 7, sets in with branch B(4), resp. B(7), having a root of order 3 4 , resp. 3 7 , in Figure 1 and 2, resp. 3 and 4. The metabelian skeletons of the Figures 3 and 4 were drawn by Nebelung [37, p. 189 ff], the complete trees were given by Ascione and coworkers [38], [39, Fig. 4.8, p. 76, and Fig. 6.1, p. 123].
We define two kinds of arithmetically structured graphs G of finite p-groups by mapping each vertex V ∈ G of the graph to statistical number theoretic information, e.g. the distribution of second p-class groups M = G 2 p K or p-class tower groups G = G ∞ p K, with respect to a given kind of number fields K, for instance real quadratic fields Definition 6.2. Let p be a prime and G be a subgraph of a descendant tree T of finite p-groups.
• The mapping • For an assigned upper bound B > 0, the mapping For both mappings, the subset of the graph G consisting of vertices V with MD(V ) = ∞, resp. AF(V ) = 0, is called the support of the distribution. The trivial values outside of the support will be ignored in the sequel.
Whereas  Table 2 shows the designation of the transfer kernel type [37], the IPOD κ 1 G, and the iterated multi-layered IPAD of second order, , for 3-groups G of maximal class up to order |G| = 3 8 , characterized by the logarithmic order, lo, i.e. lo(G) := log 3 |G|, and the SmallGroup identifier, id [25,26].
The groups in Table 2 are represented by vertices of the tree diagrams in Figure 1  Our extension in 1991 [41] merely produced further examples for these occurrences of type a. In the 15 years from 1991 to 2006 we consequently were convinced that this type with at least three total 3-principalizations is the only possible type of real quadratic fields.
The absolute frequencies in [5, Tbl. 2, p. 496] and [24, Tbl. 6.1, p. 451], which should be corrected by the Corrigenda in the Appendix, and the extended statistics in Figure 1 underpin the striking dominance of type a. The distribution of the second 3-class groups M = G 2 3 K with the smallest order 3 4 , resp. 3 6 , alone reaches 79.7% for the accumulated types a.2 and a.3 together, resp. 6.4% for type a.  Table 2. IPOD κ 1 G and iterated IPAD τ The most extensive computation of data concerning unramified cyclic cubic extensions L/K of the 481 756 real quadratic fields K = Q( √ d) with discriminant 0 < d < 10 9 and 3-class rank ̺ 3 K = 2 has been achieved by M. R. Bush in 2015 [42]. In the following, we focus on the partial results for 3-class groups of type (3,3), since they extend our own results of 2010 [5,24]. In the range 0 < d < 10 9 with 415 698 fundamental discriminants d of real quadratic fields K = Q( √ d) having 3-class group of type (3,3), there exist precisely 208 236 cases (50.1%) with IPAD τ (1) Proof. The results were computed with PARI/GP [14], double-checked with MAGMA [17], and kindly communicated to us by M. R. Bush, privately [42].
For establishing the connection between IPADs and IPODs we need the following bridge.
Proof. Here, we again make use of the selection rule [29, Thm. 3.5, p. 420] that only every other branch of the tree T 1 3 2 , 2 is admissible for second 3-class groups M = G 2 3 K of (real) quadratic fields K.
According to Table 2, three (isomorphism classes of) groups G share the common IPAD τ By the Artin reciprocity law [9,11], the Artin pattern AP(K) of the field K coincides with the Artin pattern AP(M) of its second 3-class group M = G 2 3 K. Remark 7.1. The huge statistical ensembles underlying the computations of Bush [42] admit a prediction of sound and reliable tendencies in the population of the "ground state". If we compare the smaller range d < 10 7 in [5] [21], we compare the results of this most recent tour de force of computing with asymptotic densities predicted by Boston, Bush and Hajir (communicated privately and yet unpublished, similar to [13]).
with abelian maximal subgroup without abelian maximal subgroup with abelian maximal subgroup without abelian maximal subgroup As mentioned in [1], we have the following criterion for distinguishing subtypes of type a: Proof. This is essentially [1, Thm. 6.1, p. 296] but can also be seen directly by comparing the column τ 1 H with the IPAD for the rows with lo = 4 and id ∈ {7, . . . , 10} in Table 2. Here the column τ 2 H, containing the second layer of the IPAD, does not permit a distinction.
Unfortunately, we also must state a negative result: Proof. This is a consequence of comparing both columns τ 1 H and τ 2 H for the rows with lo ∈ {6, 8} and id ∈ {95, . . . , 101}, resp. id ∈ {2221, . . . , 2227} in Table 2. According to the selection rule [29, Thm. 3.5, p. 420], only every other branch of the tree T 1 3 2 , 2 is admissible for second 3-class groups M = G 2 3 K of (real) quadratic fields K. Proof. Let G be a 3-group of maximal class. Then G is metabelian by [29,Thm. 3.7,proof,p. 421] or directly by [43,Thm. 6,p. 26]. Suppose that H is a non-metabelian 3-group of derived length dl(H) ≥ 3 such that H/H ′′ ≃ G. According to [4,Thm. 5.4], the Artin patterns AP(H) and AP(G) coincide, in particular, both groups share a common IPOD κ 1 H = κ 1 G, which contains at least three total kernels, indicated by zeros, κ 1 = ( * 000) [18]. However, this is a contradiction already, since any non-metabelian 3-group, which necessarily must be of coclass at least 2, is descendant of one of the five groups 243, n with n ∈ {3, 4, 6, 8, 9} whose IPODs possess at most two total kernels, and a descendant cannot have an IPOD with more total kernels than its parent, by [4,Thm. 5.2]. Consequently, the cover cov(G) of G in the sense of [23, Dfn. 5.1] consists of the single element G.
Finally, we apply this result to class field theory: Since M = G 2 3 K is assumed to be of coclass cc(M) = 1, we obtain G = G ∞ 3 K ∈ cov(M) = {M} and the length of the 3-class tower is given by Remark 7.2. To the very best of our knowledge, Theorem 7.3 does not appear in the literature, although we are convinced that it is well known to experts, since it can also be proved purely group theoretically with the aid of a theorem by Blackburn [43,Thm. 4,p. 26]. Here we prefer to give a new proof which uses the structure of descendant trees.   Table 3 shows the designation of the transfer kernel type [6,37], the IPOD κ 1 G, and the iterated multi-layered IPAD of 2nd order, for 3-groups G on the coclass tree T 2 3 5 , 6 up to order |G| = 3 8 , characterized by the logarithmic order, lo, and the SmallGroup identifier, id, [25,26]. To enable a brief reference for relative identifiers we put Q := 3 6 , 49 , since this group was called the non-CF group Q by Ascione [38,39].
The range 0 < d < 10 7 of fundamental discriminants d of real quadratic fields K = Q( √ d) of type E, which underlies Theorem 7.8 in this section, resp. 7.12 in the next section, is just sufficient to prove that each of the possible groups G in Theorem 7.5, resp. 7.9, is actually realized by the 3-tower group G ∞ 3 K of some field K.
with IPOD of type E.6 or E.14 for 0 < d < 10 7 [5], [24].) In the range 0 < d < 10 7 of fundamental discriminants d of real quadratic fields K = Q( √ d), there exist precisely 3, resp. 4, cases with 3-principalization type E.6, κ 1 K ∼ (1313), resp. E.14, Proof. The results of [24, Tbl. 6.5, p. 452], where the entry in the last column freq. should be 28 instead of 29 in the first row and 4 instead of 3 in the fourth row, were computed in 2010 by means of the free number theoretic computer algebra system PARI/GP [14] using an implementation of our own principalization algorithm in a PARI script, as described in detail in [24, § 5, pp. 446-450]. The accumulated frequency 7 for the second and third row was recently split into 3 and 4 with the aid of the computational algebra system MAGMA [17]. See also [5,Tbl. 4,p. 498].
Remark 7.3. The minimal discriminant d = 5 264 069 of real quadratic fields K = Q( √ d) of type E.6, resp. d = 3 918 837 of type E.14, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group G 2 3 K, on the branch B(6) of the coclass tree T 2 243, 6 in Figure 3. Remark 7.4. The computation of the 3-principalization type E.14 of the field with d = 9 433 849 resisted all attempts with MAGMA versions up to V2.21-7. Due to essential improvements in the change from relative to absolute number fields, made by the staff of the Computational Algebra Group at the University of Sydney, it actually became feasible to figure it out with V2.21-8 [17] for UNIX/LINUX machines or V2.22-3 for any operating system. 7.6. 3-groups G of coclass cc(G) = 2 arising from 3 5 , 8 . Table 4 shows the designation of the transfer kernel type, the IPOD κ 1 G, and the iterated multi-layered IPAD of second order, τ (2) * G = [τ 0 G; [τ 0 H; τ 1 H; τ 2 H] H∈Lyr 1 G ], for 3-groups G on the coclass tree T 2 3 5 , 8 up to order |G| = 3 8 , characterized by the logarithmic order, lo, and the SmallGroup identifier, id [25,26]. To enable a brief reference for relative identifiers we put U := 3 6 , 54 , since this group was called the non-CF group U by Ascione [38,39].
The groups in Table 4 are represented by vertices of the tree diagram in Figure 4. Table 4. IPOD κ 1 G and iterated IPAD τ  Proof. This is essentially [1,Thm. 6.3,. It is also an immediate consequence of Table  4, which has been computed with MAGMA [17]. As a termination criterion we can now use the more precise [4,Thm. 5      Theorem 7.10. In dependence on the parameters c, t and k, the IPAD of second order of G has the form where a variant B(3, n) of the nearly homocyclic abelian 3-group A(3, n) of order n ≥ 2 is defined as in Formula (7.4) of Theorem 7.6.
7.8. Number fields with IPOD of type E.8 or E.9. Let K be a number field with 3-class group Cl 3 K ≃ C 3 × C 3 and first layer Lyr 1 K = {L 1 , . . . , L 4 } of unramified abelian extensions.
Remark 7.5. The minimal discriminant d = 6 098 360 of real quadratic fields K = Q( √ d) of type E.8, resp. d = 342 664 of type E.9, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group G 2 3 K, on the branch B(6) of the coclass tree T 2 243, 8 in Figure 4. Remark 7.6. The 3-principalization type E.9 of the field with d = 9 674 841 could not be computed with MAGMA versions up to V2.21-7. Finally, we succeeded to figure it out by means of V2.21-8 [17].  Table 5 shows the designation of the transfer kernel type, the IPOD κ 1 G, and the iterated multi-layered IPAD of second order, τ (2) * G = [τ 0 G; [τ 0 H; τ 1 H; τ 2 H] H∈Lyr 1 G ], for sporadic 3-groups G of type H.4 up to order |G| = 3 8 , characterized by the logarithmic order, lo, and the SmallGroup identifier, id [25,26]. To enable a brief reference for relative identifiers we put N := 3 6 , 45 , since this group was called the non-CF group N by Ascione [38,39].
The groups in Table 5 are represented by vertices of the tree diagram in Figure 6. Figure 6 visualizes sporadic 3-groups of section § 7.9 which arise as 3-class tower groups G = G ∞ 3 K of real quadratic fields K = Q( √ d), d > 0, with 3-principalization type H.4 and the corresponding minimal discriminants, resp. absolute frequencies, in Theorem 7.13 and 7.14.
The tree is infinite, according to Bartholdi,Bush [45] and  2 187 3 7  Proof. Extensions of absolute degrees 6 and 18 were constructed in steps with MAGMA [17], using the class field package of C. Fieker [46]. The resulting iterated IPADs of second order τ (2) K were used for the identification, according to Table 5, which is also contained in the more extensive theorem [1,Thm. 6.5,.   [47, pp. 80-83]. In 1989 already, we recognized that only for the discriminant d = −21 668 one of the four absolute cubic subfields L i , 1 ≤ i ≤ 4, of the unramified cyclic cubic extensions N i of K has 3-class number h 3 L i = 9, which is not the case for the other 6 cases of type H.4 in the table [47, pp. 78-79]. According to [5,Prop. 4.4,p. 485] or [5,Thm. 4.2,p. 489] or [48], the exceptional cubic field L i is contained in a sextic field N i with 3-class number Remark 7.8. The imaginary quadratic field with discriminant d = −21 668 possesses the same 3principalization type H.4, but its second 3-class group G 2 3 K is isomorphic to either 3 7 , 286 −#1; 2 or 3 7 , 287 − #1; 2 of order 3 8 , and has the different IPAD τ (1) K = [1 2 ; 32, 1 3 , (21) 2 ]. Results for this field will be given in [20]. Proof. Using the technique of Fieker [46], extensions of absolute degrees 6 and 54 were constructed in two steps, squeezing MAGMA [17] close to its limits. The resulting multi-layered iterated IPADs of second order τ (2) * K were used for the identification, according to Table 5, resp. the more detailed theorem [1,Thm. 6.5,. Table 6 shows the designation of the transfer kernel type, the IPOD κ 1 G, and the iterated multi-layered IPAD of second order, for sporadic 3-groups G of type G.19 up to order |G| = 3 14 , characterized by the logarithmic order, lo, and the SmallGroup identifier, id [25,26], resp. the relative identifier for lo ≥ 9. To enable a brief reference for relative identifiers we put W := 3 6 , 57 , since this group was called the non-CF group W by Ascione [39,38] The groups in Table 6 are represented by vertices of the tree diagram in Figure 7. Figure 7 visualizes sporadic 3-groups of Table 6 which arise as 3-class tower groups G = G ∞ 3 K of real quadratic fields K = Q( √ d), d > 0, with 3-principalization type G.19 and the corresponding minimal discriminants, resp. absolute frequencies in Theorem 7.16 and 7.17.
For d = +24 126 593, −12 067 and −54 195, we can only give the conjectural location of G. Since real quadratic fields of type G.19 seemed to have a very rigid behaviour with respect to their 3-class field tower, admitting no variation at all, we were curious about the continuation of these discriminants beyond the range d < 10 7 . Fortunately, M. R. Bush granted access to his extended numerical results for d < 10 9 [42], and so we are able to state the following unexpected answer to our question "Is the 3-class tower group G of real quadratic fields with type G. 19   Proof. Similar to the proof of Theorem 7.15, using Table 6, but now applied to the more extensive range of discriminants and various iterated IPADs of second order.  Proof. The results of [24,Tbl. 6.4,p. 452] were computed in 2010 by means of PARI/GP [14] using an implementation of our principalization algorithm, as described in [24, § 5, pp. 446-450]. The frequency 94 in the last column "freq." of the first row concerns type G.19 in the bigger range −10 6 < d < 0. Reduced to the first half of this range, we have 46 occurrences. Proof. Similar to the proof of Theorem 7.15, using Table 6, but now applied to the different range of discriminants and various iterated IPADs of second order. Meanwhile we succeeded in computing the second layer of the transfer target type, τ 2 K, for the three critical fields with the aid of the computational algebra system MAGMA [17] by determining the structure of the 3-class groups Cl 3 L of the 13 unramified bicyclic bicubic extensions L/K with relative degree [L : K] = 3 2 and absolute degree 18. In accumulated (unordered) form the second layer of the TTT is given by τ 2 K = [32 5  These results admit incredibly powerful conclusions, which bring us closer to the ultimate goal to determine the precise isomorphism type of G 2 3 K. Firstly, they clearly show that the second 3-class groups of the three critical fields are pairwise non-isomorphic without using the IPODs. Secondly, the component with the biggest order establishes an impressively sharpened estimate for the order of G 2 3 K from below. The background is explained by the following lemma. Lemma 8.1. Let G be a finite p-group with abelianization G/G ′ of type (p, p, p) and denote by lo p (G) := log p (ord(G)) the logarithmic order of G with respect to the prime number p. Then the abelianizations H/H ′ of subgroups H < G in various layers of G admit lower bounds for lo p (G): (1) lo p (G) ≥ 1 + max{lo p (H/H ′ ) | H ∈ Lyr 1 G}.
(2) lo p (G) ≥ 2 + max{lo p (H/H ′ ) | H ∈ Lyr 2 G}. The assumption that a maximal subgroup U < M having not the biggest order of U/U ′ were abelian (with U/U ′ ≃ U ) immediately yields the contradiction that Proof. As mentioned earlier already, computations with MAGMA [17] have shown that the accumulated second layer of the TTT is given by τ 2 K = [32 5  Unfortunately, it was impossible for any of the three critical fields K to compute the third layer of the TTT, τ 3 K, that is the structure of the 3-class group of the Hilbert 3-class field F 1 3 K of K, which is of absolute degree 54. This would have given the precise order of the metabelian group M = G 2 3 K = Gal(F 2 3 K/K), according to Lemma 8.1, since M ′ = Gal(F 2 3 K/F 1 3 K) ≃ Cl 3 (F 1 3 K). We also investigated whether the complete iterated IPAD of second order, τ (2) M, is able to improve the lower bounds in Theorem 8.2 further. It turned out that, firstly none of the additional non-normal components of (τ 1 H) H∈Lyr 1 M seems to have bigger order than the normal components of τ 2 M, and secondly, due to the huge 3-ranks of the involved groups, the number of required class group computations enters astronomic regions.
To give an impression, we show the results for five of the 13 maximal subgroups in the case of

Acknowledgements
We gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008-N25.
Sincere thanks are given to Michael R. Bush (Washington and Lee University, Lexington, VA) for making available numerical results on IPADs of real quadratic fields K = Q( √ d), and the distribution of discriminants d < 10 9 over these IPADs [42].
We are indebted to Nigel Boston, Michael R. Bush and Farshid Hajir for kindly making available an unpublished database containing numerical results of their paper [13] and a related paper on real quadratic fields, which is still in preparation.
A succinct version of the present article has been delivered on July 09, 2015, within the frame of the 29ièmes Journées Arithmeétiques at the University of Debrecen, Hungary [2].
(1) The restriction of the numerical results in Proposition 7.1 to the range 0 < d < 10 7 is in perfect accordance with our machine calculations by means of PARI/GP [14] in 2010, thus providing the first independent verification of data in [5,24,29]. However, in the manual evaluation of this extensive data material for the ground state of the types a.1, a.2, a.3, and a.3 * , a few errors crept in, which must be corrected at three locations: in the tables [ The absolute frequency of the ground state is actually given by 1 382 instead of the incorrect 1 386 for the union of types a.2 and a.3, 698 instead of the incorrect 697 for type a.3 * , 2 080 instead of the incorrect 2 083 for the union of types a.2, a.3, and a.3 * , and