Deep transfers of p-class tower groups

Let p be a prime. For any finite p-group G, the deep transfers T(H,G'):H/H' -->G'/G'' from the maximal subgroups H of index (G:H)=p in G to the derived subgroup G' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels kappa_d(G)=(ker(T(H,G')))_{(G:H)=p}. For all finite 3-groups G of coclass cc(G)=1, the family kappa_d(G) is determined explicitly. The results are applied to the Galois groups G=Gal(F_3^\infty/F) of the Hilbert 3-class towers of all real quadratic fields F=Q(d^1/2) with fundamental discriminants d>1, 3-class group Cl_3(F)~C_3*C_3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1


Introduction
The layout of this paper is the following. Deep transfers of finite p-groups G, with an assigned prime number p, are introduced as an innovative supplement to the (usual) shallow transfers [8] in § 2. The family κ d (G) = (ker(T H,G ′ )) (G:H)=p of the kernels of all deep transfers of G is called the deep transfer kernel type of G and will play a crucial role in this paper. For all finite 3-groups G of coclass cc(G) = 1, the deep transfer kernel type κ d (G) = (ker(T Hi,G ′ )) 1≤i≤4 is determined explicitly with the aid of commutator calculus in § 3 using a parametrized polycyclic power-commutator presentation of G [3,22,23]. In the concluding § 4, the orders of the deep transfer kernels are sufficient for identifying the Galois group G ∞ 3 F := Gal(F We recall [8] that the sTKT is usually simplified by a family of non-negative integers, in the following way. For 1 ≤ i ≤ p + 1, (2.2) κ s (G) i := j if ker(T G,Hi ) = H j /G ′ for some j ∈ {1, . . . , p + 1}, 0 if ker(T G,Hi ) = G/G ′ .
The progressive innovation in this paper, however, is the introduction of the deep Artin transfer.
Definition 2.2. By the deep transfers we understand the Artin transfer homomorphisms T Hi,G ′ : [15] from the maximal subgroups H 1 , . . . , H p+1 to the commutator subgroup G ′ of G, which forms the deep layer Lyr 2 (G) of the (unique) subgroup of index (G : G ′ ) = p 2 of G with abelian quotient G/G ′ . Accordingly, we call the family (2.3) κ d (G) = (# ker(T Hi,G ′ )) 1≤i≤p+1 the deep transfer kernel type (dTKT) of G.
We point out that, as opposed to the sTKT, the members of the dTKT are only cardinalities, since this will suffice for reaching our intended goals in this paper. This preliminary coarse definition is open to further refinement in subsequent publications. (See the proof of Theorem 3.1.)

Identification of 3-groups by deep transfers
The drawback of the sTKT is the fact that occasionally several non-isomorphic p-groups G share a common Artin pattern AP(G) := (τ (G), κ s (G)) [21,Thm. 7.2,p. 158]. The benefit of the dTKT is its ability to distinguish the members of such batches of p-groups which have been inseparable up to now. After the general introduction of the dTKT for arbitrary p-groups in § 2, we are now going to demonstrate its advantages in the particular situation of the prime p = 3 and finite 3-groups G of coclass cc(G) = 1, which are necessarily metabelian with second derived subgroup G ′′ = 1 and abelianization G/G ′ ≃ C 3 × C 3 , according to Blackburn [2]. For the statement of our main theorem, we need a precise ordering of the four maximal subgroups H 1 , . . . , H 4 of the group G = x, y , which can be generated by two elements x, y, according to the Burnside basis theorem. For this purpose, we select the generators x, y such that and H 1 = χ 2 (G), provided that G is of nilpotency class cl(G) ≥ 3. Here we denote by the two-step centralizer of G ′ in G, where we let (γ i (G)) i≥1 be the lower central series of G =: The identification of the groups will be achieved with the aid of parametrized polycyclic powercommutator presentations, as given by Blackburn [3], Miech [22], and Nebelung [23]: , 1} and w, z ∈ {−1, 0, 1} are bounded parameters, and the index of nilpotency n = cl(G) + 1 = cl(G) + cc(G) = log 3 (ord(G)) =: lo(G) is an unbounded parameter. Lemma 3.1. Let G be an arbitrary group with elements x, y ∈ G. Then the second and third power of the product xy are given by (1) , then (xy) 2 = x 2 y 2 s 2 s −a n−1 and (xy) 3 = x 3 y 3 s 3 2 s 3 s −2a n−1 , and the second and third power of xy 2 are given by (xy 2 ) 2 = x 2 y 4 s 2 2 s −2a n−1 and (xy 2 ) 3 = x 3 y 6 s 6 2 s 2 3 s −2a n−1 . Proof. We prepare the calculation of the powers by proving a few preliminary identities: yx = 1 · yx = xyy −1 x −1 · yx = xy · y −1 x −1 yx = xy · [y, x] = xys 2 , and similarly s 2 y = ys 2 · [s 2 , y] = ys 2 t 3 and t 3 y = yt 3 · [t 3 , y] = yt 3 t 4 and s 2 x = xs 2 · [s 2 , x] = xs 2 s 3 and s 3 y = ys 3 · [s 3 , y] = ys 3 u 4 and u 4 y = yu 4 · [u 4 , y] = yu 4 u 5 . Furthermore, yx 2 = yx · x = xys 2 · x = xy · s 2 x = xy · xs 2 s 3 = x · yx · s 2 s 3 = x · xys 2 · s 2 s 3 = x 2 ys 2 2 s 3 , s 2 y 2 = s 2 y · y = ys 2 t 3 · y = ys 2 · t 3 y = ys 2 · yt 3 t 4 = y · s 2 y · t 3 t 4 = y · ys 2 t 3 · t 3 t 4 = y 2 s 2 t 2 3 t 4 , s 3 y 2 = s 3 y · y = ys 3 u 4 · y = ys 3 · u 4 y = ys 3 · yu 4 u 5 = y · s 3 y · u 4 u 5 = y · ys 3 u 4 · u 4 u 5 = y 2 s 3 u 2 4 u 5 . Now the second power of xy is (xy) 2 = xyxy = x · yx · y = x · xys 2 · y = x 2 y · s 2 y = x 2 y · ys 2 t 3 = x 2 y 2 s 2 t 3 and the third power of xy is (xy) 3 = xy · (xy) 2 = xy · x 2 y 2 s 2 t 3 = x · yx 2 · y 2 s 2 t 3 = x · x 2 ys 2 2 s 3 · y 2 s 2 t 3 = x 3 ys 2 2 · s 3 y 2 · s 2 t 3 = = x 3 ys 2 2 · y 2 s 3 u 2 4 u 5 · s 2 t 3 = x 3 ys 2 · s 2 y 2 · s 3 u 2 4 u 5 , then t 4 = u 4 = u 5 = 1, t 3 = s −a n−1 , t 3 3 = s −3a n−1 = 1, and G ′ is abelian. Then the shallow and deep transfer kernel type of G are given in dependence on the relational parameters a, n, w, z of G ≃ G n a (z, w) by Table 1.  (9,9,9,9) Proof. The shallow TKT κ s (G) of all 3-groups G of coclass cc(G) = 1 has been determined in [8], where the designations a.n of the types were introduced with n ∈ {1, 2, 3}. Here, we indicate a capable mainline vertex of the tree Fig. 1-2, pp. 142-143] by the type a.1 * with a trailing asterisk. As usual, type a.3 * indicates the unique 3-group . For this purpose, we need expressions for the images of the deep Artin transfers T i := T Hi, First, we consider the distinguished two-step centralizer The second kernel equation Thus, the deep transfer kernel is given by independently of a, n, w, z. Consequently, the deep transfer kernel is given by For the inner transfer, we have Therefore, the deep transfer kernel is given by Thus, the deep transfer kernel is given by These finer results are summarized in terms of coarser cardinalities in Table 1. 4. Arithmetical application to 3-class tower groups 4.1. Real quadratic fields. As a final highlight of our progressive innovations, we come to a number theoretic application of Theorem 3.1, more precisely, the unambiguous identification of the pro-3 Galois group G ∞ 3 F = Gal(F The first field of this kind with d = 62 501 was discovered by Heider and Schmithals in 1982 [4]. They computed the sTKT κ s (F ) = (0, 0, 0, 0) with four total 3-principalizations in the unramified cyclic cubic extensions E i /F , 1 ≤ i ≤ 4, on a CDC Cyber mainframe. The fact that d = 62 501 is a triadic irregular discriminant (in the sense of Gauss) with non-cyclic 3-class group Cl 3 (F ) ≃ C 3 ×C 3 has been pointed out earlier in 1936 by Pall [24] already. The second field of this kind with d = 152 949 was discovered by ourselves in 1991 by computing κ s (F ) on an AMDAHL mainframe [6]. In 2006, there followed d = 252 977 and d = 358 285, and many other cases in 2009 [7,11].
In both tables, the shortcut MD means the minimal discriminant [21, Dfn. 6.2, p. 148].  The diagram in Figure 1 visualizes the initial eight branches of the coclass tree T 1 (R) with abelian root R = 9, 2 ≃ C 3 × C 3 . Basic definitions, facts, and notation concerning general descendant trees of finite p-groups are summarized briefly in [10, § 2, pp. 410-411], [9]. They are discussed thoroughly in the broadest detail in the initial sections of [12]. Descendant trees are crucial for recent progress in the theory of p-class field towers [16,19,20], in particular for describing the mutual location of the second p-class group G 2 p F and the p-class tower group G ∞ p F of a number field F . Generally, the vertices of the coclass tree in the figure 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) = n − 1 denotes the nilpotency class of G, and |G| = 3|H|, that is, γ c (G) ≃ C 3 is cyclic of order 3. See also [10, § 2.2, p. 410-411] and [12, § 4, p. 163-164].
The vertices of the tree diagram in Figure 1 are classified by using various symbols: (1) big contour squares represent abelian groups, (2) big full discs • represent metabelian groups with at least one abelian maximal subgroup, (3) small full discs • represent metabelian groups without abelian maximal subgroups.
with abelian maximal subgroup without abelian maximal subgroup The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library [1] of MAGMA [5]. We omit the orders, which are given on the left hand scale. The sTKT κ s [8, Thm. 2.5, Tbl. [6][7], in the bottom rectangle concerns all vertices located vertically above. The first component τ (1) of the TTT [13, Dfn. 3.3, p. 288] in the left rectangle concerns vertices G on the same horizontal level containing an abelian maximal subgroup. It is given in logarithmic notation. The periodicity with length 2 of branches, B(j) ≃ B(j + 2) for j ≥ 4, sets in with branch B(4), having a root of order 3 4 . 3-class tower groups G = G ∞ 3 F with coclass cc(G) = 1 of real quadratic fields F = Q( √ d) are located as arithmetically realized vertices on the tree diagram in Figure 1. The minimal fundamental discriminants d, i.e. the MDs, are indicated by underlined boldface integers adjacent to the oval surrounding the realized vertex [1,5,15].

4.2.
Totally real dihedral fields. In fact, we have computed much more information with MAGMA than mentioned at the end of the previous section 4.1. To understand the actual scope of our numerical results it is necessary to recall that each unramified cyclic cubic relative extension E i /F , 1 ≤ i ≤ 4, gives rise to a dihedral absolute extension E i /Q of degree 6, that is an S 3extension [7,Prp. 4.1,p. 482]. For the trailing three fields E i , 2 ≤ i ≤ 4, in the stable part of the TTT τ (F ) = [ (9,9), (3, 3) 3 ], i.e. with Cl 3 (E i ) of type (3, 3), we have constructed the unramified cyclic cubic extensionsẼ i,j /E i , 1 ≤ j ≤ 4, and determined the Artin pattern AP(E i ) of E i , in particular, the 3-principalization type of E i in the fieldsẼ i,j . The dihedral fields E i of degree 6 share a common polarizationẼ i,1 = F (1) 3 , the Hilbert 3-class field of F , which is contained in the relative 3-genus field (E i /F ) * , whereas the other extensionsẼ i,j with 2 ≤ j ≤ 4 are non-abelian over F , for each 2 ≤ i ≤ 4. Our computational results suggest the following conjecture concerning the infinite family of totally real dihedral fields E i for varying real quadratic fields F . . Let E 2 , E 3 , E 4 be the three unramified cyclic cubic relative extensions of F with 3-class group Cl 3 (E i ) of type (3,3).
Then E i /Q is a totally real dihedral extension of degree 6, for each 2 ≤ i ≤ 4, and the connection between the component κ d (F ) i = # ker(T F (1) 3 /Ei ) of the deep transfer kernel type κ d (F ) of F and the 3-class tower group G i = G ∞ 3 E i = Gal((E i ) (∞) 3 /E i ) of E i is given in the following way:   Table 4. A provable argument for the truth of the conjecture is the fact that κ d (F ) i = # ker(T F (1) 3 /Ei ) = #κ s (E i ) 1 = #κ s (G i ) 1 , for 2 ≤ i ≤ 4, but it does not explain why the sTKT κ s (G i ) is a.2 with a fixed point if κ d (F ) i = 3. It is interesting that a dihedral field E i of degree 6 is satisfied with a non-σ group, such as 243, 27 , as its 3-class tower group. On the other hand, it is not surprising that a mainline group, such as 243, 26 with sTKT a.1 * and relation rank d 2 = 4, is possible as G i = G ∞ 3 E i , since the upper Shafarevich bound for the relation rank of the 3-class tower group of a totally real dihedral field E i of degree 6 with Cl 3 (E i ) ≃ C 3 × C 3 is given by ̺ + r 1 + r 2 − 1 = 2 + 6 + 0 − 1 = 7 > 4 [19, Thm. 1.3, p. 75].

Acknowledgements
The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): P 26008-N25.