Index-\(p\) abelianization data of \(p\)-class tower groups

Given a fixed prime number \(p\), the multiplet of abelian type invariants of the \(p\)-class groups of all unramified cyclic degree \(p\) extensions of a number field \(K\) is called its IPAD (index-\(p\) abelianization data). These invariants have proved to be a valuable information for determining the Galois group \(G_p^2\) of the second Hilbert \(p\)-class field and the \(p\)-capitulation type \(\varkappa\) of \(K\). For \(p=3\) and a number field \(K\) with elementary \(p\)-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group \(G_p^\infty\) of the maximal unramified pro-\(p\) extension of \(K\).


Introduction
After a thorough discussion of the terminology used in this article, the logarithmic and power form of abelian type invariants in § 2, and multilayered transfer target types (TTTs), ordered and accumulated index-p abelianization data (IPADs) up to the third order in § 3, we state the main results in § 3.1 on IPADs of exceptional form, and in § 3.2 on IPADs in parametrized infinite sequences. These main theorems give all possible IPADs of number fields K with 3-class group Cl 3 (K) of type (3,3).
Before we turn to applications in extreme computing, that is, squeezing the computational algebra systems PARI [33] and MAGMA [6,7,23] to their limits in § 5, where we show how to detect malformed IPADs in § 5.1, and how to complete partial p-capitulation types in § 5.2, we have to establish a componentwise correspondence between transfer kernel types (TKTs) and IPADs in § 4 by exploiting details of proofs which were given in [28].
Iterated IPADs of second order are used in § 6 for the indirect calculation of TKTs in § 6.1, and for determining the exact length ℓ p (K) of the p-class tower of a number field K in § 6.2. This sophisticated technique proves ℓ 3 (K) = 3 for K = Q( √ d) with d ∈ {342 664, 957 013} (the first real quadratic fields) and d = −3 896 (the first tough complex quadratic field after the 'easy' d = −9 748 [13]), which resisted all attempts up to now.
Finally, we emphasize that infinite p-class towers admit an unknown wealth of possible fine structure in § 7 on complex quadratic fields K having a 3-class group Cl 3 (K) of type (3, 3, 3).

Abelian type invariants
Let p be a prime number and A be a finite abelian p-group. According to the main theorem on finitely generated abelian groups, there exists a non-negative integer r ≥ 0, the rank of A, and a sequence n 1 , . . . , n r of positive integers such that n 1 ≤ n 2 ≤ . . . ≤ n r and (2.1) A ≃ Z/p n1 Z ⊕ . . . ⊕ Z/p nr Z.
The cumbersome subscripts can be avoided by defining m j := n r1+...+rj for each 1 ≤ j ≤ s. Then (1) For abelian type invariants of high complexity, the logarithmic form in Definition 2.1 requires considerably less space (e.g. in tables) than the usual power form (2) For brevity, we can even omit the commas separating the entries of the logarithmic form of abelian type invariants, provided all the m j remain smaller than 10. (3) A further advantage of the brief logarithmic notation is the independence of the prime p, in particular when p-groups with distinct p are being compared. (4) Finally, since our preference is to select generators of finite p-groups with decreasing orders, we agree to write abelian type invariants from the right to the left, in both forms. Now let 0 ≤ n ≤ v be a fixed integer and suppose that K ≤ L ≤ F 1 p (K) belongs to the n-th layer. Then the Galois group H = Gal(F ∞ p (K) | L) is of finite index (G : H) = [L : K] = p n in the p-tower group G of K and the quotient G/H ≃ Gal(L | K) is abelian, since H contains the commutator subgroup G ′ = Gal(F ∞ p (K) | F 1 p (K)) of G. Definition 3.2. For each integer 0 ≤ n ≤ v, the system A further application of Artin's reciprocity law [1] shows that , for every subgroup H ∈ Lyr n (G) and its corresponding extension field L ∈ Lyr n (K), where 0 ≤ n ≤ v is fixed (but arbitrary).
Since   Similarly, the multiplet τ n (K) = (Cl p (L)) L∈Lyr n (K) , where each member Cl p (L) is interpreted rather as its abelian type invariants, is called the n-th layer of the transfer target type (TTT) of the number field K, , where τ n (K) = (Cl p (L)) L∈Lyr n (K) for each 0 ≤ n ≤ v.
(1) If it is necessary to specify the underlying prime number p, then the symbol τ (p, G), resp. τ (p, K), can be used for the TTT.
(2) Suppose that 0 < n < v. If an ordering is defined for the elements of Lyr n (G), resp. Lyr n (K), then the same ordering is applied to the members of the layer τ n (G), resp. τ n (K), and the TTT layer is called ordered. Otherwise, the TTT layer is called unordered or accumulated, since equal components are collected in powers with formal exponents denoting iteration. (3) In view of the considerations in Equation (3.3), it is clear that we have the equality in the sense of componentwise isomorphisms.
Since it is increasingly difficult to compute the structure of the p-class groups Cl p (L) of extension fields L ∈ Lyr n (K) in higher layers with n ≥ 2, it is frequently sufficient to make use of information in the first layer only, that is the layer of subgroups with index p. Therefore, Boston, Bush and Hajir [8] invented the following first order approximation of the TTT, a concept which had been used in earlier work already [9,12,3,10,31], without explicit terminology.
of the TTT τ (G), resp. τ (K), to the zeroth and first layer is called the index-p abelianization data (IPAD) of G, resp. K.
So, the complete TTT is an extension of the IPAD. However, there also exists another extension of the IPAD which is not covered by the TTT. It has also been used already in previous investigations by Boston, Bush and Nover [12,10,31] and is constructed from the usual IPAD [τ 0 (K); τ 1 (K)] of K, firstly, by observing that τ 1 (K) = (Cl p (L)) L∈Lyr 1 (K) can be viewed as τ 1 (K) = (τ 0 (L)) L∈Lyr 1 (K) and, secondly, by extending each τ 0 (L) to the IPAD [τ 0 (L); τ 1 (L)] of L.
Definition 3.5. The family is called the iterated IPAD of second order of G, resp. K.
The concept of iterated IPADs as given in Dfn. 3.5 is restricted to the second order and first layers, and thus is open for further generalization (higher orders and higher layers). Since it could be useful for 2-power extensions, whose absolute degrees increase moderately and remain manageable by MAGMA or PARI, we briefly indicate how the iterated IPAD of third order could be defined: 3.1. Sporadic IPADs. In the next two central theorems, we present complete specifications of all possible IPADs of pro-p groups G for p = 3 and the simplest case of an abelianization G/G ′ of type (3,3). We start with pro-3-groups G whose metabelianizations G/G ′′ are vertices on sporadic parts of coclass graphs outside of coclass trees. Since the abelian type invariants of the members of TTT layers will depend on the parity of the nilpotency class c or coclass r, a more economic notation, avoiding the tedious distinction of the cases odd or even, is provided by the following definition. Definition 3.6. For an integer n ≥ 2, the nearly homocyclic abelian 3-group A(3, n) of order 3 n is defined by its type invariants (q + r, q)=(3 q+r , 3 q ), where the quotient q ≥ 1 and the remainder 0 ≤ r < 2 are determined uniquely by the Euclidean division n = 2q + r. Two degenerate cases are included by putting A(3, 1) = (1)=(3) the cyclic group C 3 of order 3 and A(3, 0) = (0)=1 the trivial group of order 1.
The first member H 1 /H ′ 1 of the ordered first layer τ 1 (G) reveals a uni-polarization (dependence on the nilpotency class c) whereas the other three members show a stabilization (independence of c) for fixed coclass r.
Proof. Again, we make use of [28], and we point out that, for a p-group G, the index of nilpotency m = c + 1 is used generally instead of the nilpotency class cl(G) = c = m − 1 and the invariant e = r + 1 frequently (but not always) replaces the coclass cc(G) = r = e − 1 in that paper.
(1) All components of τ 1 (G) are given in [28,  For coclass bigger than 2, it is irrelevant to which of the four (in the case of odd coclass r) or six (in the case of even coclass r) coclass trees the group G belongs. The IPAD is independent of this detailed information, provided that c ≥ r + 3.

Componentwise correspondence of IPAD and TKT
Within this section, where generally p = 3, we employ some special terminology. We say a class of a base field K remains resistant if it does not capitulate in any unramified cyclic cubic extension L|K. When the 3-class group of K is of type (3, 3) the next layer of unramified abelian extensions is already the top layer consisting of the Hilbert 3-class field F 1 3 (K), where the resistant class must capitulate, according to the Hilbert/Artin/Furtwängler principal ideal theorem.
Our desire is to show that the components of the ordered IPAD and TKT are in a strict correspondence to each other. For this purpose, we use details of the proofs given in [28], where generators of metabelian 3-groups G with G/G ′ ≃ (3, 3) were selected in a canonical way, particularly adequate for theoretical aspects. Since we now prefer a more computational aspect, we translate the results into a form which is given by the computational algebra system MAGMA [23].
To be specific, we choose the vertices of two important coclass trees for illustrating these peculiar techniques. The vertices of depth (distance from the mainline) at most 1 of both coclass trees, with roots 243, 6   (1) polarization (dependence on the class c) at the first component, (2) stabilization (independence of the class c) at the last three components, (3) rank 3 at the second TTT component (ε = 1 in [28]). Using the class c, resp. an asterisk, as wildcard characters, these common properties can be summarized as follows, now including the details of the stabilization: However, to assure the general applicability of the theorems and corollaries in this section, we aim at independency of the selection of generators (and thus invariance under permutations).   Observe that in [28], the index of nilpotency m = c + 1 and the invariant e = r + 1 are used rather than the nilpotency class c = m − 1 and the coclass r = e − 1. The claims are a consequence of [28,§ 4.5,Tbl.4.7,p.441], when we perform a permutation from the first layer TKT and TTT with respect to the canonical generators, to the corresponding invariants with respect to MAGMA's generators.  Using the class c, resp. an asterisk, as wildcard characters, the common properties can be summarized as follows, now including details of the stabilization: Again, we have to ensure the general applicability of the following theorem and corollary, which must be independent of the choice of generators (and thus invariant under permutations).  To verify predicted asymptotic densities of maximal unramified pro-3 extensions in the article [8] numerically, the IPADs of all complex quadratic fields K = Q( √ d) with discriminants −10 8 < d < 0 and 3-class rank r 3 (K) = 2 were computed with the aid of PARI/GP [33]. In particular, there occurred 276 375, resp. 122 444, such fields with 3-class group Cl 3 (K) of type (3, 3), resp. (9, 3).  (3,3) in the range −10 8 < d < 0 of discriminants, for which Theorem 3.2 states that the 3-class groups of the 4 unramified cyclic cubic extensions can only have 3-rank 2, except for the unique type (3,3,3), revealed that the following 5 IPADs were computed erroneously by the used version of PARI/GP [33] in [8]. The successful recomputation was done with MAGMA [23].  (9,9,3), (27,9)] contained the malformed component (9,9,3) instead of the correct (9,3). This could be a TKT E.6 or E.14 or H.4. is particularly spectacular.
Example 5.2. We also checked all 122 444 IPADs for complex quadratic fields with type (9,3) in the range −10 8 < d < 0 of discriminants, Again, we found exactly 5 errors among these IPADs which had been computed by PARI/GP [33] in [8]. For the recomputation we used MAGMA [23].
The study of this extensive material was very helpful for the deeper understanding of 3-groups having abelianization of type (9,3). Systematic results in the style of Theorems 3.1 and 3.2 will be given in a forthcoming paper. The abbreviation pTKT means the punctured TKT.
Fortunately, there appeared a single discriminant only for each of the 5 erroneous IPADs, in both examples. This indicates that the errors are not systematic but rather stochastic.

Application 2:
Completing partial capitulation types. According to Theorem 3.2, the second 3-class group G of K must be of coclass cc(G) = 2, and the polarized component 54 of the IPAD shows that c − k = 5 + 4 = 9 and thus the nilpotency class c = cl(G) and the defect of commutativity k are given by either c = 9, k = 0, or c = 10, k = 1. Further, due to the lack of a rank-3 component 1 3 in the IPAD, G must be a vertex of the coclass tree T 2 ( 729, 54 ).

6.2.
Length of the p-class tower. In this section, we use the iterated IPAD of second order τ (2) (K) = [τ 0 (K); (τ 0 (L); τ 1 (L)) L∈Lyr 1 (K) ] for the indirect computation of the length ℓ p (K) of the p-class tower of a number field K, where p denotes a fixed prime.
By means of the techniques described in [13], a search in the complete descendant tree T ( 243, 6 ), not restricted to groups of coclass 2, yields exactly six candidates for the group G: three metabelian groups 2187, i with i ∈ {288, 289, 290}, and three groups of derived length 3 and order 3 8 with generalized identifiers 729, 49 − #2; i, i ∈ {4, 5, 6}. There cannot exist adequate groups of bigger orders. The former three groups are charcterized by Equations (6.4) the latter three groups (see [29, § 20.2, Fig.8]) by Equations (6.5).
Finally, we have ℓ 3 (K) = dl(G).   Fig.3.7, p.443] and has nilpotency class 3 + 2 = 5, due to the polarization. According to § 4.2, the lack of a total principalization excludes the TKT c.21 and the absence of a 2-cycle discourages the TKT G.16, whence the group G must be of TKT E.8 or E.9.
As we have shown in detail in [13], a search in the complete descendant tree T ( 243, 8 ), not restricted to groups of coclass 2, yields exactly six candidates for the group G: three metabelian groups 2187, i with i ∈ {302, 304, 306}, and three groups of derived length 3 and order 3 8 with generalized identifiers 729, 54 − #2; i, i ∈ {2, 4, 6}. There cannot exist adequate groups of bigger orders. The former three groups are characterized by Equations (6.6) the latter three groups (see [29, § 20.2, Fig.9]) by Equations (6.7).
Eventually, the 3-tower length of K, ℓ 3 (K) = dl(G), coincides with the derived length of G.
The complex quadratic analogue k = Q( √ −9 748) was known since 1934 by the famous paper of Scholz and Taussky [34]. However, it required almost 80 years until M.R. Bush and ourselves [13] succeeded in providing the first faultless proof that k has a 3-class tower of exact length ℓ 3 (k) = 3 with 3-tower group G one of the two Schur σ-groups 729, 54 − #2; i, i ∈ {2, 6}, of order 3 8 .
For K = Q( √ 342 664), the methods in [13] do not admit a final decision about the length ℓ 3 (K). They only yield four possible 3-tower groups of K, namely either the two unbalanced groups 2187, i with i ∈ {302, 306} and relation rank r = 3 bigger than the generator rank d = 2 or the two Schur σ-groups 729, 54 − #2; i with i ∈ {2, 6} and r = 2 equal to d = 2.
As a final coronation of this section, we show that our new IPAD strategies are powerful enough to enable the determination of the length ℓ 3 (K) with the aid of information on the structure of 3-class groups of number fields of absolute degree 6 · 9 = 54.
For this purpose, we extend the concept of iterated IPADs of second order (1) If the IPAD of K is given by then the first layer TKT is κ 1 (K) = (4, 1, 1, 1) and there exist infinitely many possibilities ℓ 3 (K) ≥ 2 for the length of the 3-class tower of K.

Complex quadratic fields of 3-rank three
In this concluding section we present another impressive application of IPADs. Due to Koch and Venkov [22], it is known that a complex quadratic field K with 3-class rank r 3 (K) ≥ 3 has an infinite 3-class field tower K < F 1 3 (K) < F 2 3 (K) < . . . < F ∞ 3 (K) of length ℓ 3 (K) = ∞. In the time between 1973 and 1978, Diaz y Diaz [14,15] and Buell [11] have determined the smallest absolute discriminants |d| of such fields. Recently, we have launched a computational project which aims at verifying these classical results and adding sophisticated arithmetical details. Below the bound 10 7 there exist 25 discriminants d of this kind, and 14 of the corresponding fields K have a 3-class group Cl 3 (K) of elementary abelian type (3,3,3). For each of these 14 fields, we determine the type of 3-principalization κ := κ 1 (3, K) in the thirteen unramified cyclic cubic extensions L 1 , . . . , L 13 of K, and the structure of the 3-class groups Cl 3 (L i ) of these extensions, i.e., the IPAD of K. We characterize the metabelian Galois group G = G 2 3 (K) = Gal(F 2 3 (K)|K) of the second Hilbert 3-class field F 2 3 (K) by means of kernels and targets of its Artin transfer homomorphisms [2] to maximal subgroups. We provide evidence of a wealth of structure in the set of infinite topological 3-class field tower groups G ∞ 3 (K) = Gal(F ∞ 3 (K)|K) by showing that the 14 groups G are pairwise non-isomorphic.
We summarize our results and their obvious conclusion in the following theorem.
Theorem 7.1. There exist exactly 14 complex quadratic number fields K = Q( √ d) with 3-class groups Cl 3 (K) of type (3,3,3) and discriminants in the range −10 7 < d < 0. They have pairwise non-isomorphic (1) second and higher 3-class groups Gal(F n 3 (K)|K), n ≥ 2, (2) infinite topological 3-class field tower groups Gal(F ∞ 3 (K)|K). Before we come to the proof of Theorem 7.1 in § 7.3, we collect basic numerical data concerning fields with r 3 (K) = 3 in § 7.1, and we completely determine sophisticated arithmetical invariants in § 7.2 for all fields with Cl 3 (K) of type (3, 3, 3). The first attempt to do so for the smallest absolute discriminant |d| = 3 321 607 with r 3 (K) = 3 is due to Heider and Schmithals in [20, § 4, Tbl.2, p.18], but it resulted in partial success only. 7.1. Discriminants −10 7 < d < 0 of fields K = Q( √ d) with rank r 3 (K) = 3. Since one of our aims is to investigate tendencies for the coclass of second and higher p-class groups G n p (K) = Gal(F n p (K)|K), n ≥ 2, [25,27] of a series of algebraic number fields K with infinite p-class field tower, for an odd prime p ≥ 3, the most obvious choice which suggests itself is to take the smallest possible prime p = 3 and to select complex quadratic fields K = Q( √ d), d < 0, having the simplest possible 3-class group Cl 3 (K) of rank 3, that is, of elementary abelian type (3,3,3).
The reason is that Koch and Venkov [22] have improved the lower bound of Golod, Shafarevich [35,19] and Vinberg [39] for the p-class rank r p (K), which ensures an infinite p-class tower of a complex quadratic field K, from 4 to 3.
To provide an independent verification, we used the computational algebra system Magma [6,7,23] for compiling a list of all quadratic fundamental discriminants −10 7 < d < 0 of fields K = Q( √ d) with 3-class rank r 3 (K) = 3. In 16 hours of CPU time we obtained the 25 desired discriminants and the abelian type invariants (here written in 3-power form) of the corresponding 3-class groups Cl 3 (K), and also of the complete class groups Cl(K), as given in Table 1 There are 14 discriminants, starting with d = −4 447 704, such that Cl 3 (K) is elementary abelian of type (3,3,3), and 10 discriminants, starting with −3 321 607, such that Cl 3 (K) is of non-elementary type (9,3,3). For the single discriminant d = −5 153 431, we have a 3-class group of type (27,3,3). We have published this information in the Online Encyclopedia of Integer Sequences (OEIS) [37], sequences A244574 and A244575.  give 3-logarithms of abelian type invariants and we denote iteration by formal exponents. Note that the multiplets κ and τ are ordered and in componentwise mutual correspondence, in the sense of § 4. In Table 4, we classify each of the 14 complex quadratic fields K = Q( √ d) of type (3, 3, 3) according to the occupation numbers of the abelian type invariants of the 3-class groups Cl 3 (L i ) of the 13 unramified cyclic cubic extensions L i , that is the accumulated (unordered) form of the IPAD of K. Whereas the dominant part of these groups is of order 3 6 = 729, there always exist(s) at least one and at most four distinguished groups of bigger order, usually 3 8 = 6 561 and occasionally even 3 10 = 59 049, According to the number of distinguished groups, we speak about uni-, bi-, tri-or tetra-polarization. If the maximal value of the order is 3 8 , then we have a ground state, otherwise an excited state.