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 [[1], p. 50], [[2], 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 [3] [4] [5]. 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
(2.1)
containing the targets and kernels of the Artin transfer homomorphisms
[[5], Lem. 6.4, p. 198], [[2], 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], §2.2, pp. 475-476] that the sTKT is usually simplified by a family of non-negative integers, in the following way. For
,
(2.2)
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
[[5], Lem. 6.1, p. 196], [[6], 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
(2.3)
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
[[7], 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 [8].
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
(3.1)
and
, provided that G is of nilpotency class
. Here we denote by
(3.2)
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 [3], Miech [4], and Nebelung [5]:
(3.3)
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 [2], where the designations a.n of the types were introduced with
. Here, we indicate a capable mainline vertex of the tree
with root
[7] 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 [8].) Generally, we have to distinguish outer transfers,
if
[[2], Equation (4), p. 470], and inner transfers,
if
and h is selected in
[[2], Equation (6), p. 486].
First, we consider the distinguished two-step centralizer
with
. Then
and
if
(
abelian), but
if
(
non-abelian) [[2], 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
(3.4)
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
(3.5)
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
(3.6)
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
(3.7)
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 [9]. 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 [10] already. The second field of this kind with
was discovered by ourselves in 1991 by computing
on an AMDAHL mainframe [11]. In 2006, there followed
and
, and many other cases in 2009 [12] [13].
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:
(4.1)
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 [1] [14], 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 [2], 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
[[15], 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 [[12], 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 [12] [13] [16] [17] 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 [[7], 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 [[18], §2, pp. 410-411] [19]. They are discussed thoroughly in the broadest detail in the initial sections of [20]. Descendant trees are crucial for recent progress in the theory of p-class field towers [15] [21] [22], 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 [[18], §2.2, p. 410-411] and [[20], §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 [23] [24] of MAGMA [25]. We omit the orders, which are given on the left hand scale. The sTKT
[[2] Thm. 2.5, Tbl. 6-7], in the bottom rectangle concerns all vertices located vertically above. The first component
of the TTT [[26] [27], 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 [6] [24] [25].
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 [[6], Dfn. 5.5, p. 84]. The computations were done with the aid of the computational algebra system MAGMA [25]. 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 [[6], Dfn. 5.5, p. 84]. For more details on the concepts stabilization and polarization, see [[6], §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 [28] [29], 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 [[12], 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:
(4.2)
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
[[15], 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).
Acknowledgements
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 [30].