First, we recall the concepts of abelian type invariants and abelian quotient invariants in the context of finite p-groups and infinite pro-p groups, and we specify our conventions in their notation.

Let p ≥ 2 be a prime number. It is well known that a finite abelian group A with order | A | a power of p possesses a unique representation

A ≃ ⊕ i = 1 s ( ℤ / p e i ℤ ) r i (2.1)

as a direct sum with integers s ≥ 0 , r i ≥ 1 for 1 ≤ i ≤ s , and strictly decreasing e 1 > ⋯ > e s ≥ 1 .

Definition 2.1 The abelian type invariants of A are given either in power form,

ATI ( A ) : = [ p e 1 , ⋯ , p e 1 ︷ r 1   times , ⋯ , p e s , ⋯ , p e s ︷ r s   times ] , (2.2)

or in logarithmic form with formal exponents indicating iteration,

ATI ( A ) : = [ e 1 r 1 , ⋯ , e s r s ] . (2.3)

Let G be a pro-p group with commutator subgroup G ′ and finite abelianization G a b : = G / G ′ .

Definition 2.2 The abelian quotient invariants of G are the abelian type invariants of the biggest abelian quotient of G

AQI ( G ) : = ATI ( G a b ) . (2.4)

Next, we explain the concept of transfer kernel type of finite p-groups and infinite pro-p groups.

Due to our assumptions, the first layer Lyr 1 G of subgroups of G is a finite set consisting of maximal normal subgroups S of G with abelian quotients G / S . Consequently, the Artin transfer homomorphism from G to S ∈ Lyr 1 G is distinguished by a very simple mapping law:

T G , S : G / G ′ → S / S ′ ,   g ⋅ G ′ ↦ ( g p ⋅ S ′ if   g ∈ ( G / G ′ ) \ ( S / G ′ ) , g 1 + h + h 2 + ⋯ + h p − 1 ⋅ S ′ if   g ∈ S / G ′ , (3.1)

where h denotes an arbitrary element in G \ S ( [1] , 4.1, p. 76).

The Artin limit pattern encapsulates particular group theoretic information (connected with Artin transfers) about the lattice of subgroups of G, where each element U has at least one predecessor, except the root G itself. We select a unique predecessor in the following way: for U ∈ Lyr 1 S we put π ( U ) : = S , and we add the formal definition π ( G ) : = G . This enables a recursive construction, as follows:

Definition 3.1 The collection of Artin transfers up to order n of G is defined recursively by

α ( 1 ) G : = T π ( G ) , G     and     α ( n ) G : = [ α ( 1 ) G ;   ( α ( n − 1 ) S ) S ∈ Lyr 1 G ]     for   n ≥ 2. (3.2)

The limit of this infinite recursive nesting process is denoted by

α ( ∞ ) G : = l i m n → ∞ α ( n ) G (3.3)

and is called the Artin transfer collection of G.

Remark 3.1 By means of the collection of Artin transfers up to order three,

α ( 3 ) G = [ T G , G ; ( α ( 2 ) S ) S ∈ Lyr 1 G ] = [ T G , G ; ( [ T G , S ; ( T S , U ) U ∈ Lyr 1 S ] ) S ∈ Lyr 1 G ] ,

it should be emphasized that our definition of stepwise relative mappings T G , S and T S , U admits finer information than the corresponding absolute mappings T G , U = T S , U ∘ T G , S ( [1] , Thm. 3.3, p. 72), since in general the kernel of T S , U cannot be reconstructed from T G , U and T G , S .

The infinite collection of mappings α ( ∞ ) G is only the foundation for the objects τ ( ∞ ) G and ϰ ( ∞ ) G we are really interested in.

Definition 3.2 The iterated abelian quotient invariants up to order n of G are defined recursively by

τ ( 1 ) G : = AQI ( G )     and     τ ( n ) G : = [ τ ( 1 ) G ; ( τ ( n − 1 ) S ) S ∈ Lyr 1 G ]     for   n ≥ 2. (3.4)

Similarly, the iterated transfer kernels up to order n of G are defined recursively by

ϰ ( 1 ) G : = k e r ( T π ( G ) , G )     and     ϰ ( n ) G : = [ ϰ ( 1 ) G ; ( ϰ ( n − 1 ) S ) S ∈ Lyr 1 G ]     for   n ≥ 2. (3.5)

Both are collected in the nth order Artin pattern AP ( n ) G : = ( τ ( n ) G , ϰ ( n ) G ) of G. The limits of these infinite recursive nesting processes are called the abelian invariant collection of G,

τ ( ∞ ) G : = l i m n → ∞ τ ( n ) G , (3.6)

and the transfer kernel collection of G,

ϰ ( ∞ ) G : = l i m n → ∞ ϰ ( n ) G . (3.7)

Finally, the pair ALP ( G ) : = ( τ ( ∞ ) G , ϰ ( ∞ ) G ) is called the Artin limit pattern of G.

Remark 3.2 For a finite p-group G, the recursive nesting processes in the definition of the Artin limit pattern are actually finite.

The abelian quotient invariants are a unary concept, since τ ( 1 ) G = AQI ( G ) = ATI ( G / G ′ ) depends on G only. The first order abelian quotient invariants τ ( 1 ) G already contain non-trivial information on the abelianization of G.

The transfer kernels are a binary concept for S < G , since ϰ ( 1 ) S = k e r ( T π ( S ) , S ) depends on π ( S ) and S. The first order transfer kernel of G is trivial: ϰ ( 1 ) G = k e r ( T π ( G ) , G ) = k e r ( T G , G ) = k e r ( id G / G ′ ) = 1 , and non-trivial information starts with the transfer kernels of second order ϰ ( 1 ) S = k e r ( T π ( S ) , S ) = k e r ( T G , S ) for S ∈ Lyr 1 G which are members of ϰ ( 2 ) G .

The analogous constructions for a number field F instead of a pro-p group G, along the lines of §§2.1.2 and 2.2.2, lead to the Artin limit pattern ALP ( F ) : = ( τ ( ∞ ) F , ϰ ( ∞ ) F ) of F.

Let F p ( ∞ ) be the Hilbert p-class tower of the number field F, that is, the maximal unramified pro-p extension of F, and denote by G p ∞ F = Gal ( F p ( ∞ ) / F ) its Galois group, which is briefly called the p-tower group of F. Now we are going to employ the abelian type invariant collection τ ( ∞ ) F of F, and the abelian quotient invariant collection τ ( ∞ ) ( G p ∞ F ) of G p ∞ F , i.e., the first component of the respective Artin limit pattern. The transfer kernel collections ϰ ( ∞ ) will be considered further in §5.

Theorem 3.1 For each integer n ≥ 1 , the abelian quotient invariants of nth order of the p-tower group G p ∞ F of F are equal to the abelian type invariants of nth order of the number field F

( ∀ n ≥ 1 )   τ ( n ) ( G p ∞ F ) = τ ( n ) F     and   thus   τ ( ∞ ) ( G p ∞ F ) = τ ( ∞ ) F . (3.8)

The invariant type of the p-tower group G p ∞ F of F coincides with the invariant type of the number field F

ϰ ( G p ∞ F ) S p + 1 = ϰ ( F ) S p + 1 . (3.9)

Even the orbit representatives of the transfer kernel types of G p ∞ F and F coincide,

ϰ ( G p ∞ F ) = ( k e r ( T G p ∞ F , U i ) ) 1 ≤ i ≤ p + 1 = ( k e r ( T F , E i ) ) 1 ≤ i ≤ p + 1 = ϰ ( F ) , (3.10)

provided that the U i ∈ Lyr 1 ( G p ∞ F ) and the E i ∈ Lyr 1 F are connected by U i = Gal ( F p ( ∞ ) / E i ) , for each 1 ≤ i ≤ p + 1 . Otherwise, we only have equivalence ϰ ( G p ∞ F ) ~ ϰ ( F ) .

Proof. The claims are well-known consequences of the Artin reciprocity law of class field theory [2] [3] .

In contrast to the full p-tower group G = G p ∞ F , the Galois groups G p m F : = Gal ( F p ( m ) / F ) ≃ G / G ( m ) of the finite stages F p ( m ) of the p-class tower of F, that is, of the higher Hilbert p-class fields of the number field F, in general fail to reveal the abelian type invariants of nth order of the number field F. More precisely, there is a strict upper bound on the order n of the ATI of F which coincide with the AQI of order n of the mth p-class group G p m F of F with a fixed integer m ≥ 0 , namely the bound n ≤ m .

Theorem 3.2 (Successive Approximation Theorem.)

Let F be a number field, p a prime, and m,n integers. The abelian invariant collection τ ( ∞ ) F of F is approximated successively by the iterated AQI of sufficiently high p-class groups of F:

( ∀ n ≥ 1 )   ( ∀ m ≥ n )   τ ( n ) ( G p m F ) = τ ( n ) F . (3.11)

However, the transfer kernel type is a phenomenon of second order:

( ∀ m ≥ 2 )   ϰ ( G p m F ) ~ ϰ ( F ) , (3.12)

in particular, the metabelian second p-class group M : = G p 2 F ≃ G / G ″ of F is sufficient for determining the transfer kernel type of F.

Proof. This is one of the main results in ( [4] , Thm. 1.19, p. 78) and ( [5] , p. 13).

In general, the upper bound on the order n of the ATI of F in Theorem 3.2 seems to be sharp, in the following sense, where m = n − 1 .

Conjecture 3.1 (Stage Separation Criterion.)

Denote by l p F the length of the p-class tower of F, that is the derived length dl ( G p ∞ F ) of the p-tower group of F. It is determined in terms of iterated AQI of higher p-class groups of F by the following condition:

( ∀ n ≥ 1 )     l p F ≥ n   ⇔   τ ( n ) ( G p n − 1 F ) < τ ( n ) F . (3.13)

The sufficiency of the condition in Conjecture 3.1 is a proven theorem ( [5] , p. 13).

Our first attempt to find sound asymptotic tendencies in the distribution of higher non-abelian p-class groups G p n F = Gal ( F p ( n ) / F ) , with n ≥ 2 , among the finite p-groups was planned in 1991 already ( [6] , 3, Remark, p. 77). However, the insurmountable obstacles in the required computations limited the progress for twenty years. In 2012, we finally succeeded in the significant break-through of computing the second 3-class groups M = G 3 2 F , that is, the metabelianizations G / G ( 2 ) of the 3-class tower groups G = Gal ( F 3 ( ∞ ) / F ) of all 4596 quadratic fields F = ℚ ( d ) with fundamental discriminants in the remarkable range − 10 6 < d < 10 7 and elementary bicyclic 3-class group Cl 3 F ≃ C 3 × C 3 of rank two ( [7] , 6, pp. 495-499). The underlying computational techniques were based on the principalization algorithm via class group structure which we had invented in 2009 and implemented by means of the number theoretic computer algebra system PARI/GP [8] in 2010, as described in ( [9] , 5-6, pp. 446-455).

Throughout this paper, isomorphism classes of finite groups G are characterized uniquely by their identifier in the SmallGroups Database [10] [11] , which is denoted by a pair 〈 o , i 〉 consisting of the order o = ord ( G ) and a positive integer i, delimited with angle brackets. The counter 1 ≤ i ≤ N ( o ) is unique for a fixed value of the order o. In the computational algebra system MAGMA [12] [13] [14] , the upper bound N ( o ) can be obtained as return value of the function NumberOfSmallGroups(o), provided that IsInSmallGroupDatabase(o) returns true. The identifier of a given finite group G can be retrieved as return value of the function IdentifyGroup(G), provided that Can IdentifyGroup(o) returns true.

For the decision if the p-class tower of a number field F is trivial with length l p F = 0 it suffices to compute the class number h ( F ) of the field.

Theorem 4.1 (Trivial p-class tower.)

The p-class tower of a number field F is trivial, F p ( ∞ ) = F , with length l p F = 0 , if and only if the class number h ( F ) = # Cl ( F ) is not divisible by p, i. e., the p-class number is h p F = 1 .

Proof. The proof consists of a sequence of equivalent statements: The class number satisfies p ∤ h ( F ) . Û The p-valuation of h ( F ) is v p ( h ( F ) ) = 0 . Û The p-class number is # Cl p F = h p F = p v p ( h ( F ) ) = 1 . Û The p-class group

Cl p F = 1 is trivial. Û The p-class rank ρ p = d i m F p ( Cl ( F ) / Cl ( F ) p ) is equal

to zero. Û The number of unramified cyclic extensions E / F of degree p is

p ρ p − 1 p − 1 = p 0 − 1 p − 1 = 1 − 1 p − 1 = 0 . Û The maximal unramified p-extension F p ( ∞ ) of F

coincides with F. Û The Galois group G p ∞ F = Gal ( F p ( ∞ ) / F ) = Gal ( F / F ) = 1 is trivial. Û The length of the p-class tower is l p F = dl ( G p ∞ F ) = dl ( 1 ) = 0 .

Already C. F. Gauss was able to compute class numbers h ( F ) of quadratic fields F = ℚ ( d ) , at a time when the concept of class field theory was not yet coined. Nowadays, there exist extensive tables of quadratic class numbers which even contain the structures of the associated class groups Cl ( F ) . In 1998, Jacobson [15] covered all real quadratic fields with positive discriminants in the range 0 < d < 10 9 , and in 2016, Mosunov and Jacobson [16] investigated all imaginary quadratic fields with negative discriminants − 10 12 < d < 0 . Now we apply these results to class field theory.

Corollary 4.1 (Statistics for p = 3 .) The asymptotic proportion of imaginary quadratic fields F = ℚ ( d ) , with negative discriminants d < 0 , whose class number h ( F ) is, respectively is not, divisible by p = 3 is given as 43.99%, respectively 56.01%, by the heuristics of Cohen, Lenstra and Martinet. In Table 1, the approximations of these theoretical limits by relative frequencies in various ranges L < d < 0 are shown.

Proof. The heuristic asymptotic limits are given in ( [17] , 2, (1.1.c), p. 126). Their approximation by discriminants L < d < 0 with L = − 10 6 in ( [18] , Example, p. 843) and ( [6] , 2, Remark, and 3, Remark, p. 77), where 118   455 + 3   190 = 121   645 , is still rather far away from the limits. In contrast, the approximations associated with the bounds L = − 10 11 and L = − 10 12 in ( [16] , p. 2001) are very close already.

The first stage F p ( 1 ) of the p-class tower of a number field F is determined by the structure of the p-class group Cl p F of F as a finite abelian p-group. This is exactly the first order Artin pattern

AP ( 1 ) F = ( τ ( 1 ) F , ϰ ( 1 ) F ) = ( ATI ( Cl p F ) , k e r ( T F , F ) ) , (4.1)

since the trivial k e r ( T F , F ) = 1 does not contain information. However, only in the case of p-class rank one, ρ p = d i m F p ( Cl ( F ) / Cl ( F ) p ) = 1 , it is warranted that the exact length of the tower is l p F = 1 . A statistical example ( [6] , 2, Remark, p. 77) is shown in Table 2.

Theorem 4.2 A number field F with non-trivial cyclic p-class group Cl p F has an abelian p-class tower of exact length l p F = 1 , in fact, the Galois group G p ∞ F ≃ G p 1 F ≃ Cl p F is cyclic.

Proof. Suppose that Cl p F > 1 is non-trivial and cyclic. If the p-class tower had a length l p F ≥ 2 , the second p-class group M = G p 2 F would be a

non-abelian finite p-group with cyclic abelianization M / M ′ ≃ Cl p F . However, it is well known that a nilpotent group with cyclic abelianization is abelian, which contradicts the assumption of a length l p F ≥ 2 .

Remark 4.1 We interpret the computation of abelian type invariants τ ( 1 ) F of the Sylow 3-subgroup Cl 3 F of the ideal class group Cl ( F ) of a quadratic field F = ℚ ( d ) as the determination of the single-stage approximation G / G ′ ≃ G 3 1 F ≃ Cl 3 F of the 3-class tower group G = G 3 ∞ F of F. This step yields complete information about the lattice of all unramified abelian 3-extensions E / F within the Hilbert 3-class field F 3 ( 1 ) of F.

According to the Successive Approximation Theorem 3.2, the second stage F p ( 2 ) of the p-class tower of a number field F is determined by the second order Artin pattern

AP ( 2 ) F = ( τ ( 2 ) F , ϰ ( 2 ) F ) = ( [ ATI ( Cl p F ) ; ( ATI ( Cl p E ) ) E ∈ Lyr 1 F ] , [ k e r ( T F , F ) ; ( k e r ( T F , E ) ) E ∈ Lyr 1 F ] ) . (4.2)

The determination of AP ( 2 ) F for a quadratic field F with 3-class rank ρ 3 = 2 requires the computation of four 3-class groups Cl 3 E i of unramified cyclic cubic extensions E 1 , ⋯ , E 4 and of four transfer kernels k e r ( T F , E i ) .

Whereas Mosunov and Jacobson [16] were able to determine the class groups Cl ( F ) of more than 300 billion, precisely 303963550712, imaginary quadratic fields F with discriminants − 10 12 < d < 0 by parallel processes on multiple cores of a supercomputer in several years of total CPU time, it is currently definitely out of scope to compute the class groups Cl ( E i ) , 1 ≤ i ≤ 4 , for the 22757307168 unramified cyclic cubic extensions E i / F , of absolute degree six, of the 5689326792 imaginary quadratic fields F with discriminants − 10 12 < d < 0 and 3-class rank ρ 3 = 2 .

Therefore, it must not be underestimated that Boston, Bush and Hajir [19] succeeded in completing this task for the smaller range − 10 8 < d < 0 with 461925 imaginary quadratic fields F having 3-class rank ρ 3 = 2 , and 1847700 associated totally complex dihedral fields E i of degree six ( [7] , Prp. 4.1, p. 482). For this purpose the authors used the computational algebra system MAGMA [12] [13] [14] in a distributed process involving several processors with multiple cores. 276375 of these quadratic fields F have a 3-class group Cl 3 F ≃ C 3 × C 3 .

Imaginary quadratic fields F = ℚ ( d ) with negative discriminants d < 0 are the simplest number fields with respect to their unit group U F , which is a finite torsion group of Dirichlet unit rank zero. This fact has considerable consequences for their p-class tower groups, according to the Shafarevich theorem [20] , corrected in ( [21] , Thm. 5.1, p. 28), [22] .

Theorem 4.3 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist only two metabelian groups with GI-action (generator inverting action). and relation rank d 2 G = 2 (so-called Schur s-groups [23] [19] ), namely 〈 243,5 〉 and 〈 243,7 〉 .

1) These are the groups of smallest order which are admissible as 3-class tower groups G ≃ G 3 ∞ F of imaginary quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 21,21,1 3 ,21 ) ] , [ 1 ; ( 2241 ) ] ) then G 3 ∞ F ≃ 〈 243,5 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 1 3 ,21,1 3 ,21 ) ] , [ 1 ; ( 4224 ) ] ) then G 3 ∞ F ≃ 〈 243,7 〉 .

3) The actual distribution of these 3-class tower groups G among the 276375 imaginary quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants − 10 8 < d < 0 is presented in Table 3.

Proof. All finite 3-groups G with abelianization G / G ′ ≃ C 3 × C 3 are vertices of the descendant tree T ( R ) with abelian root R = 〈 9,2 〉 ≃ C 3 × C 3 . A search for metabelian vertices with relation rank d 2 G = 2 in this tree yields three hits 〈 27,4 〉 , 〈 243,5 〉 , and 〈 243,7 〉 , but only the latter two of them possess a GI-action.

The abelianization G / G ′ of a finite 3-group G which is realized as the 3-class tower group G p ∞ F of an algebraic number field F is isomorphic to the 3-class group Cl 3 F of F. When F is imaginary quadratic, it possesses signature ( r 1 , r 2 ) = ( 0,1 ) and torsionfree Dirichlet unit rank r = r 1 + r 2 − 1 = 0 . If G / G ′ ≃ Cl 3 F ≃ C 3 × C 3 , then the generator rank of G is d 1 G = 2 and the Shafarevich theorem implies bounds for the relation rank 2 = d 1 G ≤ d 2 G ≤ d 1 G + r = 2 .

The entries of Table 3 have been taken from [19] .

More recently, Boston, Bush and Hajir [24] used MAGMA [14] for computing the class groups of the 481756 real quadratic fields F having 3-class rank ρ 3 = 2 and discriminants in the range 0 < d < 10 9 , and the class groups of the 1927024 associated totally real dihedral fields E i of degree six, arising from unramified cyclic cubic extensions E i / F ( [7] , Prp. 4.1, p. 482). 415698 of these quadratic fields F have a 3-class group Cl 3 F ≃ C 3 × C 3 (415699 according to( [15] , Tbl. 7)).

Real quadratic fields F = ℚ ( d ) with positive discriminants d > 0 are the second simplest number fields with respect to their unit group U F , which is an infinite group of torsionfree Dirichlet unit rank one. Again, there are remarkable consequences for their p-tower groups, by the Shafarevich theorem ( [21] , Thm. 5.1, p. 28).

Theorem 4.4 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many

metabelian groups with RI-action and relation rank d 2 G = 3 (so-called Schur + 1 s-groups [24] ), but only three of minimal order 3⁴, namely 〈 81,7 〉 , 〈 81,8 〉 and 〈 81,10 〉 .

1) These are the groups of smallest order which are admissible as 3-class tower groups G ≃ G 3 ∞ F of real quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

2) Generally, for any number field F, these groups are determined uniquely by the second order Artin pattern.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 1 3 ,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 81,7 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 21,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 81,8 〉 .

(c) If AP ( 2 ) F = ( [ 1 2 ; ( 21,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 1000 ) ] ) then G 3 ∞ F ≃ 〈 81,10 〉 .

3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 0 < d < 10 9 is presented in Table 4. Additionally, the frequencies of the groups 〈 243,5 〉 and 〈 243,7 〉 in Theorem 4.3 are given.

Proof. A search for metabelian vertices G of minimal order with relation rank d 2 G = 3 in the descendant tree T ( R ) with abelian root R = 〈 9,2 〉 ≃ C 3 × C 3 yields three hits 〈 81,7 〉 , 〈 81,8 〉 , and 〈 81,10 〉 . All of them possess a RI-action.

The abelianization G / G ′ of a finite 3-group G which is realized as the 3-class tower group G p ∞ F of an algebraic number field F is isomorphic to the 3-class group Cl 3 F of F. When F is real quadratic, it possesses signature ( r 1 , r 2 ) = ( 2,0 ) and torsionfree Dirichlet unit rank r = r 1 + r 2 − 1 = 1 . If G / G ′ ≃ Cl 3 F ≃ C 3 × C 3 , then the generator rank of G is d 1 G = 2 and the Shafarevich theorem implies bounds for the relation rank 2 = d 1 G ≤ d 2 G ≤ d 1 G + r = 3 .

The entries of Table 4 have been taken from [24] .

In [24] , Boston, Bush and Hajir only computed the first component of the second order Artin pattern AP ( 2 ) F = ( τ ( 2 ) F , ϰ ( 2 ) F ) in Formula (4.2), that is, the abelian type invariants τ ( 2 ) F of second order of real quadratic fields F with discriminants 0 < d < 10 9 . Determining the second component ϰ ( 2 ) F , the transfer kernel type of F, is considerably harder with respect to the computational expense. Consequently, the most extensive numerical results on transfer kernels available currently, have been computed by ourselves for the smaller ranges 0 < d < 10 8 in [25] [26] , and, even computing third order Artin

patterns, for 0 < d < 10 7 in [27] [28] . With the aid of these results, we now illustrate that the transfer kernels k e r ( T F , E ) of 3-class extensions T F , E : Cl 3 F → Cl 3 E from real quadratic fields F to unramified cyclic cubic extensions E / F are capable of narrowing down the number of contestants for the 3-tower group G 3 ∞ F significantly, and thus of refining the statistics in [24] .

Corollary 4.2

1) If AP ( 2 ) F = ( [ 1 2 ; ( 32,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 1000 ) ] ) then G 3 ∞ F ≃ 〈 729,96 〉 .

2) If AP ( 2 ) F = ( [ 1 2 ; ( 32,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 2000 ) ] ) then G 3 ∞ F ≃ 〈 729, i 〉 with i ∈ { 97,98 } .

3) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,1 2 ,1 2 ,1 2 ) ] , [ 1 ; ( 0000 ) ] ) then G 3 ∞ F ≃ 〈 729, i 〉 with i ∈ { 99,100,101 } .

The actual distribution of these 3-class tower groups G among the 34631, respectively 2576, real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 0 < d < 10 8 , respectively 0 < d < 10 7 , is presented in Table 5.

According to the Successive Approximation Theorem 3.2, the third stage F p ( 3 ) of the p-class tower of a number field F is usually determined by the third order Artin pattern

AP ( 3 ) F = ( τ ( 3 ) F , ϰ ( 3 ) F ) = ( [ τ ( 1 ) F ; ( τ ( 2 ) E ) E ∈ Lyr 1 F ] , [ ϰ ( 1 ) F ; ( ϰ ( 2 ) E ) E ∈ Lyr 1 F ] ) . (4.3)

It is interesting, however, that there are extensive collections of quadratic fields F with 3-class towers of exact length l 3 F = 3 , which can be characterized by the second order Artin pattern already. We begin with imaginary quadratic fields F = ℚ ( d ) with discriminants d < 0 .

Theorem 4.5 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many non-

metabelian groups with GI-action and relation rank d 2 G = 2 (so-called Schur s-groups [19] [23] ), but only seven of minimal order 3⁸, namely 〈 6561, i 〉 with i ∈ { 606,616,617,618,620,622,624 } .

1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups G ≃ G 3 ∞ F of imaginary quadratic fields F with 3-class group Cl 3 F ≃ C 3 × C 3 .

2) Exceptionally, for an imaginary quadratic field F, the trailing six of these groups are determined by the second order Artin pattern already.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,1 3 ,21 ) ] , [ 1 ; ( 1313 ) ] ) then G 3 ∞ F ≃ 〈 6561,616 〉 .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,1 3 ,21 ) ] , [ 1 ; ( 2313 ) ] ) then G 3 ∞ F ≃ 〈 6561, i 〉 with i ∈ { 617,618 } .

(c) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,21,21 ) ] , [ 1 ; ( 1231 ) ] ) then G 3 ∞ F ≃ 〈 6561,622 〉 .

(d) If AP ( 2 ) F = ( [ 1 2 ; ( 32,21,21,21 ) ] , [ 1 ; ( 2231 ) ] ) then G 3 ∞ F ≃ 〈 6561, i 〉 with i ∈ { 620,624 } .

3) The actual distribution of these 3-class tower groups G among the 24476 imaginary quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants − 10 7 < d < 0 is presented in Table 6.

Proof. By a similar but more extensive search than in the proof of Theorem 4.3. Data for Table 6 has been computed by ourselves in June 2016 using MAGMA [14] .

Remark 4.2 It should be pointed out that items (1) and (2) of Theorem 4.5 are not valid for real quadratic fields, as documented in ( [29] , Thm. 7.8, p. 162, and Thm. 7.12, p. 165).

The group 〈 6561,606 〉 belongs to the infinite Shafarevich cover of the metabelian group 〈 729,45 〉 with respect to imaginary quadratic fields ( [30] , Cor. 6.2, p. 301), [31] . It shares a common second order Artin pattern with all other elements of the Shafarevich cover. Third order Artin patterns must be used for its identification, as shown in ( [29] , Thm. 7.14, p. 168).

Now we turn to real quadratic fields F = ℚ ( d ) with discriminants d > 0 .

Theorem 4.6 Among the finite 3-groups G with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 of rank two, there exist infinitely many non-

metabelian groups with RI-action and relation rank d 2 G = 3 (so-called Schur + 1 s-groups [24] ), but only nine of minimal order 3⁷, namely 〈 2187, i 〉 with i ∈ { 270,271,272,273,284,291,307,308,311 } .

1) These are the groups of smallest order which are admissible as non- metabelian 3-class tower groups G ≃ G 3 ∞ F of real quadratic fields F with 3- class group Cl 3 F ≃ C 3 × C 3 .

2) Exceptionally, for a real quadratic field F, four of these groups are determined by the second order Artin pattern already.

(a) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,21,1 3 ,21 ) ] , [ 1 ; ( 0313 ) ] ) then G 3 ∞ F ≃ 〈 2187, i 〉 with i ∈ { 284,291 } .

(b) If AP ( 2 ) F = ( [ 1 2 ; ( 2 2 ,21,21,21 ) ] , [ 1 ; ( 0231 ) ] ) then G 3 ∞ F ≃ 〈 2187, i 〉 with i ∈ { 307,308 } .

3) The actual distribution of these 3-class tower groups G among the 415698 real quadratic fields F = ℚ ( d ) with 3-class group Cl 3 F ≃ C 3 × C 3 and discriminants 1 < d < 10 9 is presented in Table 7.

Proof. The claims for transfer kernel type c.18, ϰ ( F ) ~ ( 0313 ) , are a consequence of ( [21] , Prp. 7.1, p. 32, Thm. 7.1, p. 33, and Rmk. 7.1, p. 35), those for type c.21, ϰ ( F ) ~ ( 0231 ) , have been proved in ( [21] , Prp. 8.1, p. 42, Thm. 8.1, p. 44, and Rmk. 8.2, p. 45). A slightly stronger result is the Main Theorem ( [21] , Thm. 2.1, p. 22).

Remark 4.3 The groups 〈 2187, i 〉 with i ∈ { 270,271,272,273 } are elements of the infinite Shafarevich cover of the metabelian group 〈 729,45 〉 with respect to real quadratic fields.

The group 〈 2187,311 〉 belongs to the infinite Shafarevich cover of the metabelian group 〈 729,57 〉 with respect to real quadratic fields.

These five groups share a common second order Artin pattern with all other elements of the relevant Shafarevich cover. Third order Artin patterns must be employed for their identification, as shown in ( [29] , Thm. 7.13, p. 167, and Thm. 7.15, p. 169).

Proposition 6.1 Let G be a finite 3-group with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of G is D.10, ϰ ( G ) ~ ( 2241 ) .

2) The abelian quotient invariants of the four maximal subgroups H 1 , ⋯ , H 4 of G are τ ( G ) ~ ( 21,21,1 3 ,21 ) .

3) The isomorphism types of the four maximal subgroups of G are H 1 ≃ H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 and H 3 ≃ 〈 3 4 ,13 〉 .

4) The group G is isomorphic to the Schur s-group 〈 3 5 ,5 〉 with relation rank d 2 = 2 .

Proof. We put G : = 〈 243 , 5 〉 and use the presentation [14]

G = 〈 x , y , s 2 , s 3 , t 3 | s 2 = [ y , x ] , s 3 = [ s 2 , x ] , t 3 = [ s 2 , y ] , x 3 = s 3 , y 3 = s 3 〉 .

Then we obtain the maximal subgroups

H 1 = 〈 y , G ′ 〉 = 〈 y , s 2 , s 3 〉 , since t 3 = [ s 2 , y ] ,

H 2 = 〈 x , G ′ 〉 = 〈 x , s 2 , t 3 〉 , since s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , G ′ 〉 = 〈 x y , s 2 , s 3 〉 , since [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , G ′ 〉 = 〈 x y 2 , s 2 , s 3 〉 , since [ s 2 , x y 2 ] = s 3 t 3 2 .

Using Lemma 5.1, and comparing to the abstract presentations [14]

〈 81 , 3 〉 = 〈 ξ , υ , σ 2 , τ | σ 2 = [ υ , ξ ] , τ = ξ 3 〉 and

〈 81 , 13 〉 = 〈 ξ , υ , ζ , σ 2 | σ 2 = [ υ , ξ ] , ξ 3 = σ 2 , υ 3 = ζ 3 = 1 〉 ,

we conclude

H 1 = 〈 y , s 2 , s 3 〉 = 〈 y , s 2 〉 ≃ 〈 81 , 3 〉 , since y 3 = s 3 ≠ [ s 2 , y ] = t 3 ,

H 2 = 〈 x , s 2 , t 3 〉 ≃ 〈 81 , 13 〉 , since x 3 = s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , s 2 , s 3 〉 = 〈 x y , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y ) 3 = t 3 2 ≠ [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , s 2 , s 3 〉 = 〈 x y 2 , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y 2 ) 3 = s 3 2 t 3 2 ≠ [ s 2 , x y 2 ] = s 3 t 3 2 .

Theorem 6.2 Let F = ℚ ( d ) be a quadratic field with elementary bicyclic 3-class group Cl 3 F ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of F is D.10, ϰ ( F ) ~ ( 2241 ) .

2) The abelian type invariants of the 3-class groups Cl 3 E i of the four unramified cyclic cubic extensions E i / F are τ ( F ) ~ ( 21,21,1 3 ,21 ) .

3) The second 3-class group G 3 2 F of F has the maximal subgroups H 1 ≃ H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 and H 3 ≃ 〈 3 4 ,13 〉 .

4) The 3-class tower group G 3 ∞ F of F is the Schur s-group 〈 3 5 ,5 〉 with relation rank d 2 = 2 .

Proof. The claims follow from Proposition 6.1 by applying the Successive Approximation Theorem 3.2 of first order.

Corollary 6.1 Let F be a quadratic field which satisfies one of the equivalent conditions in Theorem 6.2. Then the length of the 3-class tower of F is l 3 F = 2 . The four unramified cyclic cubic extensions E i / F are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank r ≥ 2 , and possess 3-class towers of length l 3 E i = 2 . More precisely, Cl 3 E 3 ≃ C 3 × C 3 × C 3 and G 3 ∞ E 3 ≃ 〈 3 4 ,13 〉 with relation rank d 2 = 5 , but Cl 3 E i ≃ C 9 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,3 〉 with relation rank d 2 = 4 for i ∈ { 1,2,4 } .

Proof. This is a consequence of Theorems 6.1 and 6.2, satisfying the Shafarevich theorem.

Proposition 6.2 Let G be a finite 3-group with elementary bicyclic abelianization G / G ′ ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of G is D.5, ϰ ( G ) ~ ( 4224 ) .

2) The abelian quotient invariants of the four maximal subgroups H 1 , ⋯ , H 4 of G are τ ( G ) ~ ( 1 3 ,21,1 3 ,21 ) .

3) The isomorphism types of the four maximal subgroups of G are H 1 ≃ H 3 ≃ 〈 3 4 ,13 〉 and H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 .

4) The group G is isomorphic to the Schur s-group 〈 3 5 ,7 〉 with relation rank d 2 = 2 .

Proof. We put G : = 〈 243 , 7 〉 and use the presentation [14]

G = 〈 x , y , s 2 , s 3 , t 3 | s 2 = [ y , x ] , s 3 = [ s 2 , x ] , t 3 = [ s 2 , y ] , x 3 = s 3 , y 3 = s 3 2 〉 .

Similarly as in Proposition 6.1, we obtain the maximal subgroups

H 1 = 〈 y , G ′ 〉 = 〈 y , s 2 , s 3 〉 , H 2 = 〈 x , G ′ 〉 = 〈 x , s 2 , t 3 〉 ,

H 3 = 〈 x y , G ′ 〉 = 〈 x y , s 2 , s 3 〉 , and H 4 = 〈 x y 2 , G ′ 〉 = 〈 x y 2 , s 2 , s 3 〉 .

Using Lemma 5.1, and comparing to the abstract presentations

〈 81 , 3 〉 = 〈 ξ , υ , σ 2 , τ | σ 2 = [ υ , ξ ] , τ = ξ 3 〉 and

〈 81 , 13 〉 = 〈 ξ , υ , ζ , σ 2 | σ 2 = [ υ , ξ ] , ξ 3 = σ 2 , υ 3 = ζ 3 = 1 〉 ,

we conclude

H 1 = 〈 y , s 2 , s 3 〉 = 〈 y , s 2 〉 ≃ 〈 81 , 3 〉 , since y 3 = s 3 2 ≠ [ s 2 , y ] = t 3 ,

H 2 = 〈 x , s 2 , t 3 〉 ≃ 〈 81 , 13 〉 , since x 3 = s 3 = [ s 2 , x ] ,

H 3 = 〈 x y , s 2 , s 3 〉 = 〈 x y , s 2 〉 ≃ 〈 81 , 3 〉 , since ( x y ) 3 = s 3 t 3 2 ≠ [ s 2 , x y ] = s 3 t 3 ,

H 4 = 〈 x y 2 , s 2 , s 3 〉 ≃ 〈 81 , 13 〉 , since ( x y 2 ) 3 = s 3 t 3 2 = [ s 2 , x y 2 ] .

Theorem 6.3 Let F = ℚ ( d ) be a quadratic field with elementary bicyclic 3-class group Cl 3 F ≃ C 3 × C 3 . Then the following conditions are equivalent:

1) The transfer kernel type of F is D.5, ϰ ( F ) ~ ( 4224 ) .

2) The abelian type invariants of the 3-class groups Cl 3 E i of the four unramified cyclic cubic extensions E i / F are τ ( F ) ~ ( 1 3 ,21,1 3 ,21 ) .

3) The second 3-class group G 3 2 F of F has the maximal subgroups H 1 ≃ H 3 ≃ 〈 3 4 ,13 〉 and H 2 ≃ H 4 ≃ 〈 3 4 ,3 〉 .

4) The 3-class tower group G 3 ∞ F of F is the Schur s-group 〈 3 5 ,7 〉 with relation rank d 2 = 2 .

Proof. The claims follow from Proposition 6.2 by applying the Successive Approximation Theorem 3.2 of first order.

Corollary 6.2 Let F be a quadratic field which satisfies one of the equivalent conditions in Theorem 6.3. Then the length of the 3-class tower of F is l 3 F = 2 . The four unramified cyclic cubic extensions E i / F are absolutely dihedral of degree 6, with torsionfree Dirichlet unit rank r ≥ 2 , and possess 3-class towers of length l 3 E i = 2 . More precisely, Cl 3 E i ≃ C 3 × C 3 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,13 〉 with relation rank d 2 = 5 for i ∈ { 1,3 } , but Cl 3 E i ≃ C 9 × C 3 and G 3 ∞ E i ≃ 〈 3 4 ,3 〉 with relation rank d 2 = 4 for i ∈ { 2,4 } .

Proof. This is a consequence of Theorems 6.1 and 6.3, satisfying the Shafarevich theorem.

We recall that a dihedral field E of degree 6 is an absolute Galois extension E / ℚ with group Gal ( E / ℚ ) = 〈 σ , τ | σ 3 = τ 2 = 1 , σ τ = τ σ − 1 〉 . It is a cyclic cubic relative extension E / F of its unique quadratic subfield F = E σ , and it contains three isomorphic, conjugate non-Galois cubic subfields L = E τ , L σ , L σ 2 . The conductor c of E / F is a nearly squarefree positive integer with special prime factors, and the discriminants satisfy the relations d E = c 4 d F 3 and d L = c 2 d F . Here, we shall always be concerned with unramified extensions, characterized by the conductor c = 1 , and thus d E = d F 3 , a perfect cube, and equal d L = d F .