1, 3-class group Cl3(F) □ C3 × C3, 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 < d < 107, and a few exceptional cases are pointed out for 1 < d < 108." /> 1, 3-class group Cl3(F) □ C3 × C3, 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 < d < 107, and a few exceptional cases are pointed out for 1 < d < 108." />
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 [, p. 50], [, Equation (4), p. 470] in §2. The family 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 , the deep transfer kernel type is determined explicitly with the aid of commutator calculus in §3 using a parametrized polycyclic power-commutator presentation of G   . In the concluding §4, the orders of the deep transfer kernels are sufficient for identifying the Galois group of the maximal unramified pro-3 extension of real quadratic fields with 3-class group , and total 3-principalization in each of their four unramified cyclic cubic extensions .
2. Shallow and Deep Transfer of p-Groups
With an assigned prime number , let G be a finite p-group. Since our focus in this paper will be on the simplest possible non-trivial situation, we assume that the abelianization of G is of elementary type with rank two. For applications in number theory, concerning p-class towers, the Artin pattern has proved to be a decisive collection of information on G.
Definition 2.1. The Artin pattern of G consists of two families
containing the targets and kernels of the Artin transfer homomorphisms [, Lem. 6.4, p. 198], [, Equation (4), p. 470] from G to its maximal subgroups with . Since the maximal subgroups form the shallow layer of subgroups of index of G, we shall call the the shallow transfers of G, and the shallow transfer kernel type (sTKT) of G.
We recall [, §2.2, pp. 475-476] that the sTKT is usually simplified by a family of non-negative integers, in the following way. For ,
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 [, Lem. 6.1, p. 196], [, Dfn. 3.3, p. 69] from the maximal subgroups to the commutator subgroup of G, which forms the deep layer of the (unique) subgroup of index of G with abelian quotient . Accordingly, we call the family
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.).
3. 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 [, 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 and finite 3-groups G of coclass , which are necessarily metabelian with second derived subgroup and abelianization , according to Blackburn .
For the statement of our main theorem, we need a precise ordering of the four maximal subgroups of the group , which can be generated by two elements , according to the Burnside basis theorem. For this purpose, we select the generators such that
and , provided that G is of nilpotency class . Here we denote by
the two-step centralizer of in G, where we let be the lower central series of with for , in particular, .
The identification of the groups will be achieved with the aid of parametrized polycyclic power-commutator presentations, as given by Blackburn , Miech , and Nebelung :
where and are bounded parameters, and the index of nilpotency is an unbounded parameter.
Lemma 3.1. Let G be an arbitrary group with elements . Then the second and third power of the product are given by
1) , where , ,
2) , where , , , .
If , then and , and the second and third power of are given by and .
Proof. We prepare the calculation of the powers by proving a few preliminary identities:
, and similarly and and and and . Furthermore, , , .
Now the second power of is
and the third power of is
If , then , , , and is abelian. □
Theorem 3.1. (3-groups G of coclass cc(G) = 1.) Let G be a finite 3-group of coclass and order with an integer exponent . Then the shallow and deep transfer kernel type of G are given in dependence on the relational parameters of by Table 1.
Proof. The shallow TKT of all 3-groups G of coclass has been determined in , where the designations a.n of the types were introduced with . Here, we indicate a capable mainline vertex of the tree with root  by the type a.1* with a trailing asterisk. As usual, type a.3* indicates the unique 3-group with . Now we want to determine the deep TKT , using the presentation of in Formula (3.3). For this purpose, we need expressions for the images of the deep Artin transfers , for each . (Observe that implies by .) Generally, we have to distinguish outer transfers, if [, Equation (4), p. 470], and inner transfers, if and h is selected in [, Equation (6), p. 486].
First, we consider the distinguished two-step centralizer with . Then and if ( abelian), but if ( non-abelian) [, Equation (3), p. 470]. The outer transfer is determined by . For the inner transfer, we have for all , but for , since
lies in the centre of G. The first kernel equation is solvable by either , where , , , or , , where , . The second kernel equation is solvable by
Table 1. Shallow and deep TKT of 3-groups G with .
either or . Thus, the deep transfer kernel is given by
Second, we put . Then and . The outer transfer is determined by . The inner transfer is given by , for all , independently of . Consequently, the deep transfer kernel is given by
Next, we put . Then and . The outer transfer is determined by . For the inner transfer, we have , for all , independently of . The first kernel equation is solvable by either or , .
Therefore, the deep transfer kernel is given by
Finally, we put i = 4. Then and . The outer transfer is determined by . The inner transfer is given by , for all , independently of . The first kernel equation is solvable by either or , .
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 of the maximal unramified pro-3 extension , that is the Hilbert 3-class field tower, of certain real quadratic fields with fundamental discriminant , 3-class group of elementary type , and shallow transfer kernel type a.1, , in its ground state with or in a higher excited state with , .
The first field of this kind with was discovered by Heider and Schmithals in 1982 . They computed the sTKT with four total 3-principalizations in the unramified cyclic cubic extensions , , on a CDC Cyber mainframe. The fact that is a triadic irregular discriminant (in the sense of Gauss) with non-cyclic 3-class group has been pointed out earlier in 1936 by Pall  already. The second field of this kind with was discovered by ourselves in 1991 by computing on an AMDAHL mainframe . In 2006, there followed and , and many other cases in 2009  .
Generally, there are three contestants for the group , for any assigned state , , and the following Main Theorem admits their identification by means of the deep transfer kernel type (See their statistical distribution at the end of Section 4.1.).
Theorem 4.1. (3-class tower groups G of coclass cc(G) = 1 and type a.1.) Let be a quadratic field with fundamental discriminant d, 3-class group , and shallow transfer kernel type a.1, .
Then F is real with , the 3-class tower group of F has coclass , and the relational parameters and of are given in dependence on the deep transfer kernel type as follows:
where we suppose that the state of type a.1 is determined by the transfer target type with .
Proof. Let be a quadratic field with 3-class group , denote by its four unramified cyclic cubic extensions and by the transfer homomorphisms of 3-classes.
If the 3-principalization is total, that is , for each , then F must be a real quadratic field with positive fundamental discriminant , since the order of the principalization kernels of an imaginary quadratic field F is bounded from above by , according to the Theorem on the Herbrand quotient of the unit groups .
By the Artin reciprocity law of class field theory  , the principalization type of the field F corresponds to the shallow transfer kernel type of the 3-class tower group of F, and the abelian type invariants of the 3-class group of F correspond to the abelian quotient invariants of G.
According to , a finite 3-group G with and must be of coclass . Table 1 shows that either of type a.1* with or of type a.1 with and .
For a real quadratic field F, the relation rank of the 3-class tower group is bounded by [, Thm. 1.3, pp. 75-76]. Consequently, G cannot be a non-abelian mainline vertex with of the coclass-1 tree with root , since all these vertices have the relation rank 4. According to [, Thm. 4.1 (1), p. 486], G cannot be the abelian root either, and we must have with and .
Now the claim is a consequence of Theorem 3.1 and Table 1. □
Table 2 shows that the ground state of the sTKT has the nice property that the smallest three discriminants already realize three different 3-class tower groups with , identified by their dTKT .
In Table 3, we see that the first excited state of the sTKT does not behave so well: although the smallest two discriminants     already realize two different 3-class tower groups with , we have to wait for the seventh occurrence until is realized, as the dTKT shows. The counter 7 is a typical example of a statistic delay.
The second excited state of the sTKT
Table 2. Deep TKT of 3-class tower groups G with .
Table 3. Deep TKT of 3-class tower groups G with .
, however, is well-behaved again: the smallest three discriminants already realize three different 3-class tower groups with , identified by their dTKT . (For logarithmic orders , no SmallGroup identifiers exist.) See Table 4.
In all tables, the shortcut MD means the minimal discriminant [, Dfn. 6.2, p. 148].
The diagram in Figure 1 visualizes the initial eight branches of the coclass tree with abelian root . Basic definitions, facts, and notation concerning general descendant trees of finite p-groups are summarized briefly in [, §2, pp. 410-411] . They are discussed thoroughly in the broadest detail in the initial sections of . Descendant trees are crucial for recent progress in the theory of p-class field towers   , in particular for describing the mutual location of the second p-class group and the p-class tower group of a number field G. 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 if H is isomorphic to the last lower central quotient , where denotes the nilpotency class of G, and , that is, is cyclic of order 3. See also [, §2.2, p. 410-411] and [, §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.
The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library   of MAGMA . We omit the orders, which are given on the left hand scale. The sTKT [ Thm. 2.5, Tbl. 6-7], in the bottom rectangle concerns all vertices located vertically above. The first component of the TTT [ , 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, for , sets in with branch , having a root of order 34.
3-class tower groups with coclass of real quadratic
Table 4. Deep TKT of 3-class tower groups G with .
Figure 1. Distribution of minimal discriminants for on the coclass-1 tree
fields 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   .
The double contour rectangle surrounds the vertices which became distinguishable by the progressive innovations in the present paper and were inseparable up to now.
In Table 5, we give the isomorphism type of the 3-class tower group of all real quadratic fields with 3-class group and shallow transfer kernel type a.1, , in its ground state , for the complete range of 150 fundamental discriminants d. It was determined by means of Theorem 4.1, applied to the results of computing the (restricted) deep transfer kernel type , consisting of the orders of the 3-principalization kernels of those unramified cyclic cubic extensions , , in the Hilbert 3-class field of F whose 3-class group is of type . These trailing three components of the TTT were called its stable part in [, Dfn. 5.5, p. 84]. The computations were done with the aid of the computational algebra system MAGMA . The 3-principalization kernel of the remaining extension with 3-class group of type does not contain essential information and can be omitted. This leading component of the TTT was called its polarized part in [, Dfn. 5.5, p. 84]. For more details on the concepts stabilization and polarization, see [, §6, pp. 90-95].
A systematic statistical evaluation of Table 5 shows that, with respect to the complete range , the group occurs most often with a clearly elevated relative frequency of 44%, whereas and share the common lower percentage of 28%, although the automorphism group of all three groups has the same order. However, the proportion for the upper bound 107 is obviously not settled yet, because there are remarkable fluctuations, as Table 6 shows. According to Boston, Bush and Hajir  , we have to expect an asymptotic limit of the proportions for .
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 , , gives rise to a dihedral absolute extension of degree 6, that is an -extension [, Prp. 4.1, p. 482]. For the trailing three fields , , in the stable part of the TTT , i.e. with of type , we have constructed the unramified cyclic cubic extensions , , and determined the Artin pattern of
Table 5. Statistics of 3-class tower groups G with .
Table 6. Proportions of 3-class tower groups with .
, in particular, the 3-principalization type of in the fields . The dihedral fields of degree 6 share a common polarization , the Hilbert 3-class field of F, which is contained in the relative 3-genus field , whereas the other extensions with are non-abelian over F, for each . Our computational results suggest the following conjecture concerning the infinite family of totally real dihedral fields for varying real quadratic fields F.
Conjecture 4.1. (3-class tower groups of totally real dihedral fields.) Let be a real quadratic field with fundamental discriminant , 3-class group , and shallow transfer kernel type a.1, , in the ground state with transfer target type . Let be the three unramified cyclic cubic relative extensions of F with 3-class group of type .
Then is a totally real dihedral extension of degree 6, for each , and the connection between the component of the deep transfer kernel type of F and the 3-class tower group of is given in the following way:
Remark 4.1. The conjecture is supported by all totally real dihedral fields which were involved in the computation of Table 5. A provable argument for the truth of the conjecture is the fact that , for , but it does not explain why the sTKT is a.2 with a fixed point if . It is interesting that a dihedral field of degree 6 is satisfied with a non-s group, such as , as its 3-class tower group. On the other hand, it is not surprising that a mainline group, such as with sTKT a.1* and relation rank , is possible as , since the upper Shafarevich bound for the relation rank of the 3-class tower group of a totally real dihedral field of degree 6 with is given by [, Thm. 1.3, p. 75].
Assuming an asymptotic limit of the proportion of the real quadratic 3-class tower groups for the ground state of sTKT a.1, we can also conjecture an asymptotic limit of the corresponding totally real dihedral 3-class tower groups , since the restricted dTKTs , , together contain three times the 9 and six times the 3 in Equation (4.2).
The author gratefully acknowledges that his research was supported by the Austrian Science Fund (FWF): P 26008-N25. Note added in proof: While this paper was under review, we succeeded in proving Conjecture 4.1with the aid of Theorems 5.1, 6.1, 6.5,on pages 676, 678, 682 in .
 Mayer, D.C. (1991) List of Discriminants of Totally Real Cubic Fields L, Arranged According to Their Multiplicities m and Conductors f. Computer Centre, Department of Computer Science, University of Manitoba, Winnipeg, Canada, Austrian Science Fund, Project Nr. J0497-PHY.
 Mayer, D.C. (2016) p-Capitulation over Number Fields with p-Class Rank Two. 2nd International Conference on Groups and Algebras (ICGA) 2016, Suzhou, Presentation delivered on July 26, 2016. https://doi.org/10.4236/jamp.2016.47135
 Mayer, D.C. (2011) The Distribution of Second p-Class Groups on Coclass Graphs. 27ièmes Journées Arithmétiques, Faculty of Math. and Informatics, Univ. of Vilnius, Lithuania, Presentation Delivered on 1 July 2011.
 Mayer, D.C. (2016) Recent Progress in Determining p-Class Field Towers. 1st International Colloquium of Algebra, Number Theory, Cryptography and Information Security (ANCI) 2016, Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah, Fès, Morocco, Invited Keynote Delivered on 12 November 2016. http://www.algebra.at/ANCI2016DCM.pdf