_{1}

Let
*p* be a prime and
*K* be a number field with non-trivial
*p*-class group Cl
_{p}
*K*. A crucial step in identifying the Galois group
G^{∞}_{p} of the maximal unramified pro-
*p* extension of
*K* is to determine its two-stage approximation M=G
^{2}
_{p}k, that is the second derived quotient M
≃G/G
^{n}. The family
τ_{1}K of abelian type invariants of the
*p*-class groups Cl
_{p}L of all unramified cyclic extensions
*L/K* of degree
*p* is called the
*index- abelianization data* (IPAD) of
*K*. It is able to specify a finite batch of contestants for the second
*p*-class group
* M* of
*K*. In this paper we introduce two different kinds of
*generalized* IPADs for obtaining more sophisticated results. The
*multi-layered* IPAD (
τ_{1}Kτ^{(2)}K) includes data on unramified abelian extensions
*L/K* of degree
*p*
^{2} and enables sharper bounds for the order of
*M* in the case
Cl
_{p}k≃(p,p,p), where current im-plementations of the
*p*-group generation algorithm fail to produce explicit contestants for
*M* , due to memory limitations. The
* iterated* IPAD of second order τ
^{(2)}K contains information on non-abelian unramified extensions
*L/K* of degree
*p*
^{2}, or even
*p*
^{3}, and admits the identification of the
*p*-class tower group
*G * for various infinite series of quadratic fields K=
Q(√
d) with Cl
_{p}K
≃(p,p) possessing a
*p*-class field tower of exact length
l
_{p}K=3 as a striking novelty.

In a previous article [

Index-p abelianization data (IPADs) are explained in §2. Our Main Theorem on three-stage towers of 3-class fields is communicated in §3. Basic definitions concerning the Artin transfer pattern [

In the last section §8 on multi-layered IPADs, it is our endeavour to point out that the rate of growth of successive derived quotients { A,D,E,F,G,H,a,b,c,d } , n , of the { 1, ⋯ ,25 } -class tower group p = 3 is still far from being known for imaginary quadratic fields K with Cl 3 K ≃ ( 3,3 ) -class rank X .n , where the criterion of Koch and Venkov [

Let 2 ≤ l 3 K ≤ 3 be a prime number. According to the Artin reciprocity law of class field theory [

by log ( G ) : = log 3 | G | if l ≥ 4 denotes the l = 2 -class rank of l = 2 [ [

for this fact is that the Galois group l = 3 of the maximal unramified abelian l ∈ { 2 , 3 } -extension c , which is called the first Hilbert l = 3 - class field of F,G,H , is isomorphic to the b,d -class group l ≥ 3 . The fields l = ∞ are contained in Cl 3 K ≃ ( 3,3 ) and each group p is of index G in p .

It was also Artin’s idea [

In particular, the structure of the AP ( G ) : = ( τ ( G ) , ϰ ( G ) ) -class groups G of all unramified cyclic extensions τ ( 1 ) G : = τ 0 G ; τ 1 G of relative degree ϰ ( 1 ) G : = ϰ 0 G ; ϰ 1 G can be interpreted as the abelian type invariants of all abelianizations p of subgroups p of index G in the second τ ( 2 ) G : = τ 0 G ; ( τ ( 1 ) H ) H ∈ Lyr 1 G -class group 2 nd , which has been dubbed the index- G abelianization data, briefly IPAD, AP c ( G ) of Lyr 0 G = { G } by Boston, Bush, and Hajir [

As we proved in the main theorem of [ [

Until very recently, the length 0 ≤ r < 2 of the n = 2 q + r -class tower

A ( 3 , 1 ) : = ( 1 ) = ^ ( 3 )

over a quadratic field C 3 with A ( 3 , 0 ) : = ( 0 ) = ^ 1 -class rank p , that is, with τ 0 ≃ ( 3,3 ) - class group G of type G / G ′ , M = G / G ″ , was an open problem. Apart from the proven impossibility of an abelian tower with c = cl ( M ) ≥ 2 [ [

The finite batch of contestants for 1 ≤ r ≤ 2 , specified by the IPAD ( T 3 , T 4 ) = ( ( A ( 3, r + 1 ) 2 ) if r = 2, M ∈ T 2 〈 243,8 〉 or r = 1 , ( 1 3 , A ( 3, r + 1 ) ) if r = 2, M ∈ T 2 〈 243,6 〉 , ( ( 1 3 ) 2 ) if r = 2, M ∈ T 2 〈 243,3 〉 or r ≥ 3. , can be narrowed down further if the τ 1 G = ( ( 1 ) 4 ) for M ≃ 〈 9,2 〉 , c = 1 , r = 1, τ 1 G = ( 1 2 , ( 2 ) 3 ) for M ≃ 〈 27,4 〉 , c = 2 , r = 1, τ 1 G = ( 1 3 , ( 1 2 ) 3 ) for M ≃ 〈 81,7 〉 , c = 3 , r = 1, τ 1 G = ( ( 1 3 ) 3 ,21 ) for M ≃ 〈 243,4 〉 , c = 3 , r = 2, τ 1 G = ( 1 3 , ( 21 ) 3 ) for M ≃ 〈 243,5 〉 , c = 3 , r = 2, τ 1 G = ( ( 1 3 ) 2 , ( 21 ) 2 ) for M ≃ 〈 243,7 〉 , c = 3 , r = 2 , τ 1 G = ( ( 21 ) 4 ) for M ≃ 〈 243,9 〉 , c = 3 , r = 2, τ 1 G = ( ( 1 3 ) 3 ,21 ) for M ≃ 〈 729,44 ⋯ 47 〉 , c = 4 , r = 2, τ 1 G = ( ( 21 ) 4 ) for M ≃ 〈 729,56 ⋯ 57 〉 , c = 4 , r = 2. -principalization type of τ 1 G is known. That is the family τ 2 G of all kernels c = cl ( M ) of c -class transfers 〈 9,2 〉 ≃ ( 3,3 ) from 〈 27,4 〉 to unramified cyclic superfields 〈 81,7 〉 ≃ Syl 3 ( A 9 ) of degree 〈 243, n 〉 over n ∈ { 4,5,7,9 } . In view of the open problem for the length of the 〈 729, n 〉 -class tower, there arose the question whether each possible n ∈ { 44, ⋯ ,47,56,57 } -principalization type 〈 27,3 〉 of a quadratic field 〈 81,8 ⋯ 10 〉 with 〈 243, n 〉 of type n ∈ { 3,6,8 } is associated with a fixed value of the tower length 〈 729, n 〉 .

For n ∈ { 34, ⋯ ,39 } and τ 2 G of type (3,3), there exist 23 distinct 3-principalization types [ [

A.1, D.5, D.10, E.6, E.8, E.9, E.14, F.7, F.11, F.12, F.13, G.16, G.19, H.4,

a.1, a.2, a.3, b.10, c.18, c.21, d.23, d.25.

In this article, we establish the last but one step for the proof of the following solution to the open problem for p = 3 and quadratic fields τ 0 = ( 1 2 ) with τ 1 .

Theorem 3.1. (Main theorem on the length of the 3-class tower for 3-class rank two)

1) For each of the 13 types of 3-principalization ( τ 1 ( i ) ) 1 ≤ i ≤ 4 with upper case letter Cnt p 2 ( τ 0 , τ 1 ) , there exists an imaginary quadratic field p , M , of that type such that τ 0 M = M / M ′ ≃ τ 0 .

2) For each of the 22 types of 3-principalization τ 1 M = ( H / H ′ ) H ∈ Lyr 1 M ≃ τ 1 , there exists a real quadratic field Cnt p 2 ( τ 0 , τ 1 ) = ∅ , τ 1 , of that type such that p = 3 .

Remark 3.1. Type τ 0 = ( 1 2 ) must be excluded for quadratic base fields τ 1 , according to [ [

Concerning the steps for the proof, we provide information in the form of

Remark 3.2. None of the types sets in with a length p . Type D behaves completely rigid with M , fixed class 3, and coclass 2. Type a is also confined to τ 1 M = ( H / H ′ ) H ∈ Lyr 1 M ≃ τ 1 but admits unbounded nilpotency class with fixed coclass 1. For type E, we have G 3 2 K with unbounded class and coclass for imaginary fields, and the unique exact dichotomy G 3 ∞ K for real fields. For type G , the length | G | ≥ 3 9 is fixed with unbounded class and coclass for real fields. The most extensive flexibility is revealed by fields of the types p and p , where

Type | Base Fields | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | imaginary | |||||||||

5 | quadratic | ||||||||||

Ref. | [ | [ | [ | [ | fields | ||||||

2 | 2 or 3 | 2 | 3 | real | |||||||

5 | quadratic | ||||||||||

Ref. | [ | [ | [ | [ | [ | [ | fields |

any finite unbounded length k ( M ) = 1 can occur with variable class and coclass. We expect that an actually infinite tower with • is impossible for M .

Let k ( M ) = 0 be a prime number and • be a pro- M group with finite abelianization k ( M ) = 1 , more precisely, assume that the commutator subgroup □ is of index G with an integer exponent □ .

Definition 4.1. For each integer G , let d 2 ( G ) ≤ 3 be the nth layer of normal subgroups of □ containing G .

Definition 4.2. For any intermediate group d 2 ( G ) ≥ 4 , we denote by n ∗ the Artin transfer homomorphism from n to ϰ 1 [ [

1) Let τ ( G ) : = [ τ 0 G ; ⋯ ; τ v G ] be the multi-layered transfer target type (TTT)

of G , where k ( G ) = 0 for each B ( j ) ≃ B ( j + 2 ) .

2) Let ϰ ( G ) : = [ ϰ 0 G ; ⋯ ; ϰ v G ] be the multi-layered transfer kernel type (TKT)

of j ≥ 7 , where B ( 4 ) for each B ( 7 ) .

Definition 4.3. The pair G is called the (restricted) Artin pattern of p .

Definition 4.4. The first order approximation τ ( 1 ) G : = [ τ 0 G ; τ 1 G ] of the TTT, resp. ϰ ( 1 ) G : = [ ϰ 0 G ; ϰ 1 G ] of the TKT, is called the index- M = G p 2 K abelianization data (IPAD), resp. index- p obstruction data (IPOD), of G = G p ∞ K .

Definition 4.5. τ ( 2 ) G : = [ τ 0 G ; ( τ ( 1 ) H ) H ∈ Lyr 1 G ] is called iterated IPAD of K = K ( d ) : = ℚ ( d )

order of d > 0 .

Remark 4.1. For the complete Artin pattern p see [ [

1) Since the 0th layer (top layer), G , consists of the group T alone, and p is the natural projection onto the commutator quotient with kernel MD : G → ℕ ∪ { ∞ } , V ↦ i n f { d | G p 2 K ( d ) ≃ V } , we usually omit the trivial top layer G and identify the IPOD B > 0 with the first layer AF : G → ℕ ∪ { 0 } , V ↦ # { d < B | G p 2 K ( d ) ≃ V } of the TKT.

2) In the case of an elementary abelianization of rank two, G , we also identify the TKT G with its first layer V , since the 2nd layer (bottom layer), MD ( V ) ≠ ∞ , consists of the commutator subgroup AF ( V ) ≠ 0 alone, and the kernel of AF is always total, that is MD , according to the principal ideal theorem [

Since the abelian type invariants of certain IPAD components of an assigned 3-group G depend on the parity of the nilpotency class c c ( G ) = 1 or coclass ϰ 1 G , a more economic notation, which avoids the tedious distinction of the cases odd or even, is provided by the following definition [ [

Definition 5.1. For an integer τ ∗ ( 2 ) G = τ 0 G ; τ 0 H ; τ 1 H ; τ 2 H H ∈ Lyr 1 G , , the nearly homocyclic abelian 3-group G of order | G | = 3 8 is defined by its type invariants lo , where the quotient lo ( G ) : = l o g 3 | G | and the remainder K = ℚ ( d ) are determined un- iquely by the Euclidean division d > 0 . Two degenerate cases are included by putting α the cyclic group L / K of order 3, and ℚ the trivial group of order 1.

In the following theorem and in the whole remainder of the article, we use the identifiers of finite 3-groups up to order 3^{8} as they are defined in the SmallGroups Library [

Theorem 5.1. (Complete classification of all IPADs with d = 62 501 [

0 < d < 10 9 (5.1)

where the polarized first component of ρ 3 K = 2 depends on the class d < 10 9 and defect 0 < d < 10 9 , the co-polarized second component increases with the coclass d , and the third and fourth component are completely stable for K = ℚ ( d ) but depend on the coclass tree containing 208 236 for 50.1 % in the following manner

τ ( 1 ) K = 1 2 ; 21 , ( 1 2 ) 3 (5.2)

Anomalies of finitely many, precisely 13, exceptional groups are summarized in the following list.

122 955 (5.3)

The polarization and the co-polarization we had in our mind when we spoke about a bi-polarization in [ [

Proof. Equations (5.1) and (5.2) are a succinct form of information which summarizes all statements about the first TTT layer 29.6 % in the formulas (19), (20) and (22) of [ [

The abelian group τ ( 1 ) K = 1 2 ; 2 2 , ( 1 2 ) 3 , the extra special group 11 780 , and the group 2.8 % do not fit into the general rules for 3-groups of coclass 1. These three groups appear in the top region of the tree diagram in the

The four sporadic groups τ ( 1 ) K = 1 2 ; 32 , ( 1 2 ) 3 with K and the six sporadic groups τ ( 1 ) K = 1 2 ; 21 , ( 1 2 ) 3 with ϰ 1 K = ( 1000 ) do not belong to any coclass-2 tree, as shown in

On the other hand, there is no need to list the groups ϰ 1 K = ( 2000 ) and K in formula (14), the groups τ ( 1 ) K = 1 2 ; 1 3 , ( 1 2 ) 3 with ϰ 1 K = ( 2000 ) in formula (15), and the groups K with τ ( 1 ) K = 1 2 ; 2 2 , ( 1 2 ) 3 in formula (16) of [ [

Remark 5.1. The reason why we exclude the second TTT layer ϰ 1 K = ( 0000 ) from Theorem 5.1, while it is part of [ [

the coclass T 1 〈 3 2 ,2 〉 and thus infinitely often.

Theorem 5.2. (Finiteness of the batch of contestants for the second M = G 3 2 K -class group K ) If G , τ ( 1 ) G = 1 2 ; 21 , ( 1 2 ) 3 , and 〈 81,8 ⋯ 10 〉 denotes an assigned family τ ( 1 ) G = 1 2 ; 1 3 , ( 1 2 ) 3 of four abelian type invariants, then the set 〈 81,7 〉 of all (isomorphism classes of) finite metabelian ϰ 1 G = ( 2000 ) -groups 〈 81,9 〉 such that

〈 81,10 〉 and ϰ 1 G = ( 1000 ) is finite.

Proof. We have τ ( 1 ) G = 1 2 ; 2 2 , ( 1 2 ) 3 , when 〈 729,99 ⋯ 101 〉 is malformed [ [

In this article, we shall frequently deal with finite 3-groups AP ( K ) of huge orders K for which no identifiers are available in the SmallGroups database [

Definition 6.1. Let M = G 3 2 K be a prime number and d < 10 7 be a finite d < 10 9 -group with nuclear rank 1382 2576 ≈ 53.6 % ↘ 208236 415698 ≈ 50.1 % [ [

150 2576 ≈ 5.8 % ↗ 26678 415698 ≈ 6.4 % (6.1)

for each 0.6 % and 1382 + 698 = 2080 .

Recall that a group with nuclear rank 208236 + 122955 = 331191 is a terminal leaf without any descendants.

All numerical results in this article have been computed by means of the computational algebra system MAGMA [

Basic definitions, facts, and notation concerning descendant trees of finite lo = 4 - groups are summarized briefly in [ [

Generally, the vertices of coclass trees in the Figures 1-4, of the sporadic part of a coclass graph in

lo | id | type | |||
---|---|---|---|---|---|

2 | 2 | a.1 | 0000 | 1^{2} | |

3 | 3 | a.1 | 0000 | 1^{2} | |

3 | 4 | A.1 | 1111 | 1^{2} | 1^{2} |

4 | 7 | a.3 | 2000 | 1^{2} | 1^{3} ^{} ^{2} |

4 | 8 | a.3 | 2000 | 1^{2} | 21 ^{2} |

4 | 9 | a.1 | 00000 | 1^{2} | 21 |

4 | 10 | a.2 | 1000 | 1^{2} | 21 |

5 | 25 | a.3 | 2000 | 1^{2} | 2^{2} |

5 5 | 26 27 | a.1 a.2 | 0000 1000 | IPAD like id 25 | |

IPAD like id 25 | |||||

5 | 28・・・30 | a.1 | 0000 | 1^{2} | 21 |

6 | 95 | a.1 | 0000 | 1^{2} | 32 |

6 6 | 96 97/98 | a.2 a.3 | 1000 2000 | IPAD like id 95 | |

IPAD like id 95 | |||||

6 | 99・・・101 | a.1 | 0000 | 1^{2} | 2^{2} |

7 | 386 | a.1 | 0000 | 1^{2} | 3^{2} |

7 7 | 387 388 | a.2 a.3 | 1000 2000 | IPAD like id 386 | |

IPAD like id 386 |

7 | 389・・・391 | a.1 | 0000 | 1^{2} | 32 |
---|---|---|---|---|---|

8 | 2221 | a.1 | 0000 | 1^{2} | 43 |

8 8 | 2222 2223/2224 | a.2 a.3 | 100 2000 | IPAD like id 2221 | |

IPAD like id 2221 | |||||

8 | 2225・・・2227 | a.1 | 0000 | 1^{2} |

central quotient G = G 3 ∞ K , where G denotes the nilpotency class of τ ( 1 ) G = τ 0 G ; τ 1 G , and either τ 0 G = 1 2 , that is, τ 1 G = ( 32 , 1 3 , ( 21 ) 2 ) is cyclic of order 3, or G , that is, ϰ 1 G = ( 1122 ) is bicyclic of type (3,3). See also [ [

The vertices of the tree diagrams in

1) big full discs ϰ 1 G = ( 3122 ) ∼ ( 4122 ) represent metabelian groups τ ( 2 ) G = τ 0 G ; ( τ 0 H i ; τ 1 H i ) 1 ≤ i ≤ 4 with defect G ,

2) small full discs H 1 , ⋯ , H 4 represent metabelian groups G with defect τ ( 1 ) H 2 = 1 3 ; 2 2 1, ( 1 3 ) 3 , ( 1 2 ) 9 .

In the Figures 3-5,

1) big full discs τ ( 1 ) H i = 21 ; 2 2 1, ( 2 1 ) 3 represent metabelian groups i ∈ { 3,4 } with bicyclic centre of type (3,3) and defect G ≃ 〈 3 7 , 2 8 8 〉 [ [

2) small full discs G ≃ 〈 3 7 , 2 8 9 〉 represent metabelian groups G ≃ 〈 3 7 , 2 9 0 〉 with cyclic centre of order 3 and defect τ ( 1 ) H 2 = 1 3 ; 2 2 1, ( 2 1 2 ) 3 , ( 1 2 ) 9 ,

3) small contour squares τ ( 1 ) H i = 21 ; 2 2 1, ( 3 1 ) 3 represent non-metabelian groups i ∈ { 3,4 } .

In the

1) big contour squares G ≃ 〈 3 8 , 6 1 6 〉 represent groups G ≃ 〈 3 8 , 6 1 7 〉 with relation rank G ≃ 〈 3 8 , 6 1 8 〉 ,

2) small contour squares τ ( 1 ) H 1 = 32 ; 2 2 1, ( 31 2 ) 3 represent groups M = G 3 2 K with relation rank K = ℚ ( d ) .

A symbol d > 0 adjacent to a vertex denotes the multiplicity of a batch of T 2 〈 3 5 , 6 〉 siblings, that is, immediate descendants sharing a common parent. The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups Library [^{4}, resp. 3^{7}, in

lo | id | type | ||
---|---|---|---|---|

5 | 6 | c.18 0122 | 1^{2} | |

6 | 48 | H.4 2122 | 1^{2} | |

6 6 6 | Q = 49 50 51 | c.18 0122 E.14 3122 E.6 1122 | IPAD like id 48 IPAD like id 48 IPAD like id 48 | |

7 | 284 | c.18 0122 | 1^{2} | |

7 | 285 | c.18 0122 | 1^{2} | |

7 7 7 | 286/287 288 289/290 | H.4 2122 E.6 1122 E.14 3122 | IPAD like id 285 IPAD like id 285 IPAD like id 285 | |

7 | 291 | c.18 0122 | 1^{2} | |

8 | 613 | c.18 0122 | 1^{2} | |

8 8 8 | 614/615 616 617/618 | H.4 2122 E.6 1122 E.14 3122 | IPAD like id 613 IPAD like id 613 IPAD like id 613 |

Nebelung [ [

We define two kinds of arithmetically structured graphs c ≤ 5 of finite l 3 K - groups by mapping each vertex c = 4 of the graph to statistical number theo- retic information, e.g. the distribution of second G 3 2 K ≃ 〈 3 6 , 51 〉 -class groups 〈 3 6 , 50 〉 or c = 5 -class tower groups G 3 2 K ≃ 〈 3 7 , 288 〉 , with respect to a given kind of number fields 〈 3 7 , 298 / 290 〉 , for instance real quadratic fields F 3 ∞ K with discriminant

lo | id | type | ||
---|---|---|---|---|

5 | 8 | c.21 2034 | 1^{2} | |

6 | 52 | G.16 2134 | 1^{2} | 2^{2} |

6 6 6 | 53 U = 54 55 | E.9 2434 c.21 2034 E.8 2234 | IPAD like id 52 IPAD like id 52 IPAD like id 52 | |

7 | 301/305 | G.16 2134 | 1^{2} | 32 |

7 7 7 | 302/306 303 304 | E.9 2334 c.21 2034 E.8 2234 | IPAD like id 301 IPAD like id 301 IPAD like id 301 | |

7 | 307 | c.21 2034 | 1^{2} | 2^{2} |

7 | 308 | c.21 2034 | 1^{2} | 2^{2} |

8 | 619/623 | G.16 2134 | 1^{2} | 32 |

8 8 8 | 620/624 621 622 | E.9 2334 c.21 2034 E.8 2234 | IPAD like id 619 IPAD like id 619 IPAD like id 619 |

G ≃ 〈 3 7 , 2 8 8 〉 .

Definition 6.2. Let l 3 K = 2 be a prime and K = ℚ ( d ) be a subgraph of a descendant tree 7 5 % of finite d ∈ { 3918837 , 8897192 , 9991432 } -groups.

・ The mapping

G ≃ 〈 3 7 , 2 8 9 〉 (6.2)

is called the distribution of minimal discriminants on G ≃ 〈 3 7 , 2 9 0 〉 .

・ For an assigned upper bound l 3 K = 2 , the mapping

25 % (6.3)

is called the distribution of absolute frequencies on d = 9433849 .

For both mappings, the subset of the graph G ≃ 〈 3 8 , 6 1 7 〉 consisting of vertices G ≃ 〈 3 8 , 6 1 8 〉 with

lo | id | type | ||
---|---|---|---|---|

5 | 4 | H.4 4111 | 1^{2} | 1^{3} |

6 | N = 45 | H.4 4111 | 1^{2} | 1^{3} |

7 | 270 | H.4 4111 | 1^{2} | 1^{3} |

7 | 271/272 | H.4 4111 | 1^{2} | 1^{3} |

7 | 273 | H.4 4111 | 1^{2} | 1^{3} ^{3} ^{3} |

8 | 605/606 | H.4 4111 | 1^{2} | 1^{3} |

G ≃ 〈 3 8 , 6 2 0 〉 , resp. G ≃ 〈 3 8 , 6 2 4 〉 , is called the support of the distribution. The trivial values outside of the support will be ignored in the sequel.

Whereas

τ ∗ ( 2 ) G = [ τ 0 G ; [ τ 0 H ; τ 1 H ; τ 2 H H ∈ Lyr 1 G ] ] , (7.1)

for 3-groups G of maximal class up to order cc ( G ) = 2 , characterized by the logari- thmic order, 〈 3 5 ,8 〉 , i.e. c : = cl ( G ) , and the SmallGroup identifier, id. [

lo | id w.r.t. | type | ||
---|---|---|---|---|

5 | 9 | G.19 2143 | 1^{2} | |

6 | G.19 2143 | 1^{2} | ||

7 | 311 | G.19 2143 | 1^{2} | 21 |

8 | 625・・・630 | G.19 2143 | 1^{2} | |

9 | G.19 2143 | 1^{2} | ||

9 | G.19 2143 | 1^{2} | ||

9 | G.19 2143 | 1^{2} | ||

9 | G.19 2143 | IPAD like id | ||

9 | G.19 2143 | IPAD like id | ||

10 | #1;1 | G.19 2143 | 1^{2} | 21 |

10 | G.19 2143 | 1^{2} | 21 | |

10 | G.19 2143 | 1^{2} | ||

11 | #2;1/2 | G.19 2143 | 1^{2} | |

11 | G.19 2143 | 1^{2} | ||

11 | G.19 2143 | 1^{2} | 21 | |

11 | G.19 2143 | 1^{2} | 21 | |

12 | G.19 2143 | 1^{2} | ||

12 | G.19 2143 | 1^{2} | 21 | |

12 | G.19 2143 | 1^{2} | 21 | |

14 | G.19 2143 | 1^{2} |

The groups in

Sound numerical investigations of real quadratic fields 0 < d < 10 7 with fundamental discriminant K = ℚ ( d ) started in 1982, when Heider and Schmithals [

notation of Nebelung [

Our extension in 1991 [

The absolute frequencies in [ [^{4}, resp. 3^{6}, alone reaches l 3 K = 3 for the accumulated types a.2 and a.3 together, resp. 3 6 % for type a.1.

So it is not astonishing that the first exception d ∈ { 4 760 877, 6 652 929, 7 358 937, 9 129 480 } without any total 3-principalizations did not show up earlier than in 2006 [ [

The most extensive computation of data concerning unramified cyclic cubic extensions G ≃ 〈 3 7 , 3 0 2 〉 of the 481,756 real quadratic fields G ≃ 〈 3 7 , 3 0 6 〉 with dis- criminant l 3 K = 2 and 3-class rank K = ℚ ( d ) has been achieved by M. R. Bush in 2015 [

Proposition 7.1. (IPADs of fields with type a up to ϰ 1 K = ( 2234 ) [

In the range 0 < d < 10^{9} with 415,698 fundamental discriminants d of real qua- dratic fields τ ( 1 ) K = 1 2 ; 3 2 , ( 21 ) 3 having 3-class group of type (3,3), there exist precisely

d ∈ { 4 760 877 , 6 652 929 , 7 358 937 , 8 632 716 , 9 129 480 } cases ( 2 nd ) with IPAD τ ( 1 ) K = [ 1 2 ; 21, ( 1 2 ) 3 ] ,

d ∈ { 342 664 , 1 452 185 , 1 787 945 , 4 861 720 , 5 976 988 , 6 098 360 , 7 100 889 , 8 079 101 , 9 674 841 } cases ( 2 nd ) with IPAD τ ( 1 ) K = [ 1 2 ; 1 3 , ( 1 2 ) 3 ] ,

□ cases ( d = 9 674 841 ) with IPAD τ ( 1 ) K = [ 1 2 ; 2 2 , ( 1 2 ) 3 ] , and

K = ℚ ( d ) cases ( d > 0 ) with IPAD τ ( 1 ) K = 1 [ 2 ; 32, ( 1 2 ) 3 ] .

Proof. The results were computed with PARI/GP [

For establishing the connection between IPADs and IPODs we need the following bridge.

Corollary 7.1. (Associated IPODs of fields with type a)

1) A real quadratic field ϰ 1 G with IPAD τ ( 1 ) K = [ 1 2 ; 21, ( 1 2 ) 3 ] has IPOD

either | G | = 3 8 of type a.2 or lo of type a.3.

2) A real quadratic field id with IPAD τ ( 1 ) K = [ 1 2 ; 1 3 , ( 1 2 ) 3 ] has IPOD

N of type a.3, more precisely a.3*, in view of the exceptional IPAD.

3) A real quadratic field G = G 3 ∞ K with IPAD τ ( 1 ) K = [ 1 2 ; 2 2 , ( 1 2 ) 3 ] has IPOD

d > 0 of type a.1.

4) A real quadratic field d = + 2 852 733 with IPAD τ ( 1 ) K = [ 1 2 ; 32, ( 1 2 ) 3 ] has IPOD

either d < 10 7 of type a.2 or 0 < d < 10 7 of type a.3.

Proof. Here, we again make use of the selection rule [ [

According to

In Theorem 7.4 we shall show that the mainline group G 3 2 K cannot occur as the second 3-class group of a real quadratic field. Among the remaining two possible groups, 〈 3 7 ,286 〉 − # 1 ; 2 has IPOD 〈 3 7 ,287 〉 − # 1 ; 2 and T 2 〈 3 5 ,6 〉 has IPOD τ ( 1 ) K = 1 2 ; 32,1 3 , ( 21 ) 2 .

The IPAD τ ( 1 ) G = [ 1 2 ; 2 2 , ( 1 2 ) 3 ] leads to three groups K = ℚ ( d ) with

IPOD 4 1 % and defect of commutativity d ∈ { 957 013 , 1 571 953 , 1 734 184 , 3 517 689 , 4 025 909 , 4 785 845, 4 945 973 , 5 562 969 , 5 562 969 , 6 318 733 , 7 762 296 , 8 070 637 } [ [

Concerning the IPAD τ ( 1 ) G = [ 1 2 ; 32, ( 1 2 ) 3 ] ,

with SmallGroup identifiers 2 9 % . The mainline group d ∈ { 2 023 845 , 4 425 229 , 6 418 369 , 6 469 817 , 6775224 , 6 895 612 , 7 123 493 , 9 419 261 } is discouraged by Theorem 7.4, G ≃ 〈 3 7 , 2 7 1 〉 has IPOD G ≃ 〈 3 7 , 2 7 2 〉 , and the two groups l 3 K = 3 have IPOD 1 9 % .

By the Artin reciprocity law [

Remark 7.1. The huge statistical ensembles underlying the computations of Bush [

〈 3 7 , 2 7 0 〉 = 〈 3 6 ,45 〉 − # 1 ; 1 by 〈 3 7 , 2 7 1 〉 = 〈 3 6 ,45 〉 − # 1 ; 2 for the union of types a.2 and

a.3,

and increases

〈 3 7 , 2 7 2 〉 = 〈 3 6 , 45 〉 − # 1 ; 3 by 〈 3 7 , 2 7 3 〉 = 〈 3 6 ,45 〉 − # 1 ; 4 for type a.3*, and

τ ( 2 ) K by □ for type a.1.

Of course, the accumulation of all types a.2, a.3, and a.3* with absolute frequencies

d > − 3 × 10 4 , resp. − 30 000 < d < 0 , shows a resultant de- crease

d by K = ℚ ( d ) .

For the union of the first excited states of types a.2 and a.3, we have a stagnation

ϰ 1 K = ( 4111 ) at the same percentage.

Unfortunately, the exact absolute frequency of the ground state of type a.2, resp. type a.3, is unknown for the extended range τ ( 1 ) K = 1 2 ; ( 1 3 ) 3 , 21 . It could be computed using Theorem 7.1. However, meanwhile we succeeded in separating all states of type a.2 and type a.3 up to G 3 2 K ≃ 〈 3 6 , 45 〉 by immediately figuring out the 3- principalization type with MAGMA V2.22-1. In [

As mentioned in [

Theorem 7.1. (The ground state of type a [ [

The second 3-class groups L i with the smallest order 3^{4} possessing type a.2 or a.3 can be separated by means of the iterated IPAD of second order

τ ( 2 ) M = [ τ 0 M ; [ τ 0 H ; τ 1 H ] H ∈ Lyr 1 M ] .

Proof. This is essentially [ [

Unfortunately, we also must state a negative result:

Theorem 7.2. (Excited states of type a [ [

Even the multi-layered IPAD τ ∗ ( 2 ) M = [ τ 0 M ; [ τ 0 H ; τ 1 H ; τ 2 H ] H ∈ Lyr 1 M ] of

second order is unable to separate the second 3-class groups 〈 3 7 ,286 〉 − # 1 ; 2 with order 3^{6} and type a.2 or a.3. It is also unable to distinguish between the three candidates for 〈 3 7 ,287 〉 − # 1 ; 2 of type a.1, and between the two candidates for τ ( 1 ) K = 1 2 ; 32 , 1 3 , ( 21 ) 2 of type a.3, both for orders d > − 3 × 10 4 .

Proof. This is a consequence of comparing both columns K = ℚ ( d ) and 5 0 % for the rows with d ∈ { − 3 896 , − 25 447 , − 27 355 } and G ≃ 〈 3 8 , 6 0 6 〉 , resp. l 3 K = 3 in

Theorem 7.3. (Two-stage 3-class towers of type a) For each (real) quadratic field τ ∗ ( 2 ) K with second 3-class group ϰ 1 G of maximal class the 3-class tower has exact length τ ∗ ( 2 ) G = τ 0 G ; τ 0 H ; τ 1 H ; τ 2 H H ∈ Lyr 1 G , .

Proof. Let G be a 3-group of maximal class. Then | G | = 3 14 is metabelian by [ [

Finally, we apply this result to class field theory: Since G = G 3 ∞ K is assumed to be of coclass K = � ( d ) , we obtain d > 0 and the length of the 3-class tower is given by T ( W − # 2 ; i ) . □

Remark 7.2. To the very best of our knowledge, Theorem 7.3 does not appear in the literature, although we are convinced that it is well known to experts, since it can also be proved purely group theoretically with the aid of a theorem by Blackburn [ [

Theorem 7.4. (The forbidden mainline of coclass 1) The mainline vertices of the coclass-1 tree cannot occur as second 3-class groups i ∈ { 1,3,5 } of (real) quadratic fields i ∈ { 2,4,6 } (of type a.1).

Proof. Since periodicity sets in with branch 3 20 in the

τ ∗ ( 2 ) G = [ τ 0 G ; [ τ 0 H ; τ 1 H ; τ 2 H ] H ∈ Lyr 1 G ] ,

for 3-groups □ on the coclass tree d < 10 7 up to order d < 10 9 , characte- rized by the logarithmic order, G , and the SmallGroup identifier, τ ( 1 ) ( K ) = 1 2 ; ( 21 ) 4 , [

The groups in

Theorem 7.5. (Smallest possible 3-tower groups 0 < d < 5 × 10 7 of type E.6 or E.14 [

If the IPOD of □ is of type E.6, d < 5 × 10 7 , resp. E.14, K = ℚ ( d ) , then the IPAD of second order τ ( 2 ) G = [ τ 0 G ; ( τ 0 H i ; τ 1 H i ) 1 ≤ i ≤ 4 ] , where the maximal subgroups of index 3 in d ∈ { 10 169 729 , 11 986 573 , 14 698 056 , 14 836 573 , 16 270 305 , 16 288 424 , 18 195 889 , 19 159 368 , 21 519 660 , 21 555 097 , 22 296 941 , 22 431 068 , 24 229 337 , 25 139 461 , 26 977 089 , 27 696 973 , 29 171 832 , 29 523 765 , 30 019 333 , 31 921 420 , 32 057 249 , 33 551 305 , 35 154 857 , 35 846 545 , 36 125 177 , 36 409 821 , 37 344 053 , 37 526 493 , 37 796 984 , 38 691 433 , 39 693 865 , 40 875 944 , 42 182 968 , 42 452 445 , 42 563 029 , 43 165 432 , 43 934 584 , 44 839 889 , 44 965 813 , 45 049 001 , 46 180 124 , 46 804 541 , 46 971 381 , 48 628 533 } are denoted by 8 6 % , determines the isomorphism type of τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) , ( 21 ; 1 4 , ( 2 1 ) 3 ) 3 in the following way:

1) τ ( 1 ) H 2 = [ 1 3 ; 2 2 1, ( 1 3 ) 3 , ( 1 2 ) 9 ] if and only if τ ( 1 ) H i = [ 21 ; 2 2 1, ( 2 1 ) 3 ] for 5 % if and only if d ∈ { 21 974 161, 22 759 557, 35 327 365 } , resp. τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 3 1 2 ) 3 ) , ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 3 or G ≃ 〈 3 8 , 6 2 9 〉 − # 1 ; 2 − # 1 ; 1 ,

2) τ ( 1 ) H 2 = [ 1 3 ; 2 2 1, ( 2 1 2 ) 3 , ( 1 2 ) 9 ] if and only if τ ( 1 ) H i = [ 21 ; 2 2 1, ( 3 1 ) 3 ] for 9 % if and only if d ∈ { 24 126 593 , 29 739 477 , 31 353 229 , 35 071 865 , 40 234 205 , 40 706 677 } , resp. τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 4 or 3 8 ,

whereas the component τ ( 1 ) H 1 = [ 32 ; 2 2 1, ( 31 2 ) 3 ] is fixed and does not admit a distinction.

Proof. This is essentially [ [

Let K = ℚ ( d ) be a descendant of coclass ϰ 1 K = ( 2143 ) of the root τ ( 1 ) K = 1 2 ; ( 21 ) 4 . Denote by − 10 6 < d < 0 the nilpotency class of □ , by d > − 5 × 10 5 the indicator of a three-stage group, and by K = ℚ ( d ) , resp 6 5 % , the defect of commutativity of d ∈ { − 12 067 , − 49 924 , − 60 099 , − 83 395 , − 86 551 , − 93 067 , − 152 355 , − 153 771 , − 161 751 , − 168 267 , − 195 080 , − 235 491 , − 243 896 , − 251 723 , − 283 523 , − 310 376 , − 316 259 , − 337 816 , − 339 459 , − 344 823 , − 350 483 , − 407 983 , − 421 483 , − 431 247 , − 433 732 , − 442 367 , − 444 543 , − 453 463 , − 458 724 , − 471 423 } itself if τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 4 , and of the metabelian parent G ≃ 〈 3 8 , 6 2 5 〉 − # 1 ; 2 − # 2 ; 1 | 2 if 3 11 .

Theorem 7.6. In dependence on the parameters l 3 K = 3 , | G | ≥ 3 14 and l 3 K ≥ 4 , the IPAD of second order of 1 5 % has the form

d ∈ { − 54 195, − 96 551, − 104 659, − 133 139, − 222 392, − 313 207, − 420 244 } (7.2)

where a variant of the nearly homocyclic abelian 3-group of order τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 2 2 1 ) 3 ) 4 in Definition 5.1, which can also be defined by 3 11 , l 3 K ∈ { 3 , 4 , ⋯ } , and

1 5 % (7.3)

is given by d ∈ { − 114 936, − 118 276, − 272 659, − 317 327, − 328 308, − 339 563, − 485 411 } and

τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 3 1 2 ) 3 ) 4 (7.4)

Let G ≃ 〈 3 8 , 6 2 9 〉 − # 1 ; 2 − # 2 ; 1 | 2 be a number field with 3-class group 3 11 and first layer l 3 K = 3 of unramified abelian extensions.

Theorem 7.7. (Criteria for d = − 91 643 .) Let the IPOD of τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 2 3 ) 3 ) 2 , ( 21 ; 1 4 , ( 3 2 1 ) 3 ) 2 be of type E.6, l 3 K ≥ 3 , resp. E.14, d = − 221 944 . If τ ( 2 ) K = 1 2 ; ( 21 ; 1 4 , ( 3 2 1 ) 3 ) 4 with l 3 K ≥ 3 , then

・ □

M = G 3 2 K for K = ℚ ( d ) ,

・ − 10 7 < d < 0

Cl 3 ( K ) for d ∈ { − 4 447 704, − 5 067 967, − 8 992 363 } .

Proof. Exemplarily, we conduct the proof for τ ( 1 ) K = τ 0 K ; τ 1 K = 1 3 ; 32 2 1 ; ( 21 4 ) 5 , ( 2 2 1 2 ) 7 . , which is the most important situation for our computational applications.

Searching for the Artin pattern ϰ 1 K = 1,2,6, ( 8 ) 6 ,9, ( 10 ) 2 ,13 with d = − 4 447 704 and ϰ 1 K = 1 , 2 , ( 3 ) 2 , ( 4 ) 2 , 6 , ( 7 ) 2 , 8 , ( 9 ) 2 , 12 , resp. d = − 5 067 967 , in the descendant tree ϰ 1 K = ( 2 ) 2 , 5 , 6 , 7 , ( 9 ) 2 , ( 10 ) 3 , ( 12 ) 3 with abelian root d = − 8 992 363 , unambiguously leads to the unique metabelian descendant

with path R ← 〈 3 3 ,3 〉 ← 〈 3 5 ,6 〉 ← 〈 3 6 ,49 〉 ← 〈 3 7 ,288 〉 = : M for type E.6, resp. two descendants Cl 3 L for type E.14. The bifurcation at the vertex L / K with nuclear rank two leads to a unique non-metabelian descendant with path R ← 〈 3 3 , 3 〉 ← 〈 3 5 , 6 〉 ← 〈 3 6 , 49 〉 ← 〈 3 8 , 616 〉 = : G for type E.6, resp. two descendants τ 2 K = 32 5 1 2 ; 4321 5 ; 2 5 1 3 , ( 3 2 21 5 ) 2 ; 2 4 1 4 , 32 2 1 5 ; ( 2 2 1 7 ) 3 , ( 2 3 1 5 ) 3 for type E.14 The cover of d = − 4 447 704 is

non-trivial but very simple, since it contains two elements τ 2 K = 3 2 2 2 1 4 ; ( 3 2 21 5 ) 3 ; 32 2 1 5 ; ( 2 3 1 5 ) 8 only. The decision whether d = − 5 067 967 and τ 2 K = 32 2 1 6 , ( 3 2 21 5 ) 3 ; 2 4 1 4 , 32 2 1 5 ; 2 2 1 7 , ( 2 3 1 5 ) 6 or d = − 8 992 363 and G 3 2 K requires the iterated IPADs of second order G 3 2 K of G and G / G ′ , which are listed in

The proof of Theorem 7.7, immediately justifies the following conclusions for lo p ( G ) ≥ 3 + lo p ( G ′ / G ″ ) .

Corollary 7.2. Under the assumptions of Theorem 7.7, the second and third 3-class groups of lo p ( G ) = 3 + lo p ( G ′ ) are given by their SmallGroups identifier [

if H ≤ G , then ord ( G ) = ( G : H ) ⋅ ord ( H ) , for type E.6, resp. lo p ( G ) = l o g p ( ( G : H ) ) + lo p ( H ) . for type E.14, and

if l o g p ( ( G : H ) ) = ( p n ) = n , then H ∈ Lyr n G for type E.6, resp. 0 ≤ n ≤ 3 for type E.14.

In the case of a 3-class tower ord ( H ) = ( H : H ′ ) ⋅ ord ( H ′ ) ≥ ( H : H ′ ) , of length lo p ( H ) = l o g p ( ( H : H ′ ) ) + lo p ( H ′ ) ≥ lo p ( H / H ′ ) , ,

if H ′ = 1 , then H for type E.6, resp. G for type E.14, and

if G ′ , then □ for type E.6, resp. M = G 3 2 K for type E.14.

The range ord ( M ) ≥ 3 9 of fundamental discriminants H < M of real quadratic fields H / H ′ of type E, which underlies Theorem 7.8 in this section, resp. 7.12 in the next section, is just sufficient to prove that each of the possible groups H ′ = 1 in Theorem 7.5, resp. 7.9, is actually realized by the 3-tower group M of some field lo 3 ( M ) = 9 .

Proposition 7.2. (Fields d ∈ { − 4 447 704, − 5 067 967, − 8 992 363 } with IPOD of type E.6 or E.14 for τ ( 1 ) K = τ 0 K ; τ 1 K = 1 3 ; 32 2 1 ; ( 21 4 ) 5 , ( 2 2 1 2 ) 7 [

Proof. The results of [ [

Remark 7.3. The minimal discriminant □ of real quadratic fields M = G 3 2 K of type E.6, resp. � of type E.14, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group lo 3 ( M ) ≥ 17 , on the branch d = − 4 447 704 of the coclass tree lo 3 ( M ) ≥ 16 in

Theorem 7.8. (3-Class towers d = − 5 067 967 with IPOD of type E.6 or E.14 for lo 3 ( M ) ≥ 15 ) Among the 3 real quadratic fields d = − 8 992 363 with IPOD of type E.6 in Proposition 7.2,

・ the 2 fields ( τ 2 K = 32 5 1 2 ; 4321 5 ; 2 5 1 3 , ( 3 2 21 5 ) 2 ; 2 4 1 4 , 32 2 1 5 ; ( 2 2 1 7 ) 3 , ( 2 3 1 5 ) 3 ) with discriminants

d = − 4 447 704

have the unique 3-class tower group τ 2 K = 3 2 2 2 1 4 ; ( 3 2 21 5 ) 3 ; 32 2 1 5 ; ( 2 3 1 5 ) 8 and 3-tower length d = − 5 067 967 ,

・ the single field ( τ 2 K = 32 2 1 6 , ( 3 2 21 5 ) 3 ; 2 4 1 4 , 32 2 1 5 ; 2 2 1 7 , ( 2 3 1 5 ) 6 ) with discriminant

d = 7153097

has the unique 3-class tower group M : = m a x { lo 3 ( H / H ′ ) | H ∈ Lyr 2 M } and 3-tower length M = lo 3 ( 32 5 1 2 ) = 3 + 5 × 2 + 2 × 1 = 15 .

Among the 4 real quadratic fields d = − 4 447 704 with IPOD of type E.14 in

Proposition 7.2,

・ the 3 fields ( M = lo 3 ( 3 2 2 2 1 4 ) = 2 × 3 + 2 × 2 + 4 ⋅ 1 = 14 ) with discriminants

d = − 5 067 967

have 3-class tower group M = lo 3 ( 32 2 1 6 ) = 3 + 2 × 2 + 6 × 1 = 13 or d = − 8 992 363 and 3-tower length lo 3 ( M ) ≥ 2 + m a x { lo 3 ( H / H ′ ) | H ∈ Lyr 2 M } = 2 + M ,

・ the single field ( M ) with discriminant

lo 3 ( M ) = 9

has 3-class tower group □ or τ 3 K and 3-tower length 3 .

Proof. Since all these real quadratic fields F 3 1 K have 3-capitulation

type M = G 3 2 K = Gal ( F 3 2 K / K ) or M ′ = Gal ( F 3 2 K / F 3 1 K ) ≃ Cl 3 ( F 3 1 K ) and τ ( 2 ) M IPAD τ ( 1 ) K = [ 1 2 ; 3 2 ,1 3 , ( 21 ) 2 ] , and the 4 fields with τ 2 M have d = − 4 447 704 IPAD

τ ( 1 ) H 1 = 2 2 1 2 ; 32 5 1 2 ; ( 2 3 1 5 ) 3 ; ( 3 2 21 2 ) 3 ; ( 321 4 ) 9 , ( 32 2 1 2 ) 24

whereas the 3 fields with τ ( 1 ) H 2 = 21 4 ; 32 5 1 2 ; 2 5 1 3 ; 2 4 1 4 ; 2 2 1 7 ; ( 31 6 ) 3 , ( 321 4 ) 33 ; ( 321 2 ) 81 have τ ( 1 ) H 3 = 2 2 1 2 ; 32 5 1 2 ; 32 2 1 5 ; ( 2 2 1 7 ) 2 ; ( 321 5 ) 3 , ( 32 2 1 3 ) 6 , ( 3 2 21 2 ) 3 , ( 32 2 1 2 ) 24 IPAD

τ ( 1 ) H 4 = 32 2 1 ; 32 5 1 2 ; 4321 5 ; ( 3 2 21 5 ) 2 ; ( 4321 3 ) 6 ; ( 431 4 ) 6 , ( 3 2 21 3 ) 6 , ( 4321 2 ) 9 , ( 3 3 1 2 ) 9

the claim is a consequence of Theorem 7.5. □

Remark 7.4. The computation of the 3-principalization type E.14 of the field with τ ( 1 ) H 5 = 2 2 1 2 ; 3 2 21 5 ; 32 2 1 5 , 2 4 1 4 ; 2 3 1 5 ; ( 321 3 ) 36 resisted all attempts with MAGMA versions up to V2.21-7. Due to essential improvements in the change from relative to absolute number fields, made by the staff of the Computational Algebra Group at the University of Sydney, it actually became feasible to figure it out with V2.21-8 [

τ ∗ ( 2 ) G = [ τ 0 G ; [ τ 0 H ; τ 1 H ; τ 2 H ] H ∈ Lyr 1 G ] ,

for 3-groups 698 2303 ≈ 30.3 % on the coclass tree 697 2303 ≈ 30.3 % up to order d = 2 747 001 , characte- rized by the logarithmic order, ϰ 1 = ( 0313 ) , and the SmallGroup identifier, ϰ 1 = ( 3313 ) [

The groups in

Theorem 7.9. (Smallest possible 3-tower groups G = G 3 ∞ K of type E.8 or E.9 [

If the IPOD of K is of type E.8, p , resp. E.9, Cl p K , then the IPAD of second order τ ( 2 ) G = [ τ 0 G ; ( τ 0 H i ; τ 1 H i ) 1 ≤ i ≤ 4 ] , where the maximal subgroups of index 3 in p are denoted by K , determines the isomorphism type of M = G p 2 K in the following way:

1) τ ( 1 ) H i = [ 21 ; 2 2 1, ( 2 1 ) 3 ] for τ 1 | K

if and only if p , resp. Cl p L or L / K ,

2) τ ( 1 ) H i = [ 21 ; 2 2 1, ( 3 1 ) 3 ] for p

if and only if K , resp. p or M ,

whereas the component τ ( 1 ) H 2 = [ 32 ; 2 2 1, ( 31 2 ) 3 ] is fixed and does not admit a distinction.

Proof. This is essentially [ [

Let Cl p K ≃ ( p , p , p ) be a descendant of coclass p of the root M . Denote by τ ( 2 ) K the nilpotency class of p 2 , by p 3 the indicator of a three-stage group, and by p , resp G , the defect of commutativity of K = ℚ ( d ) itself if Cl p K ≃ ( p , p ) , and of the metabelian parent l p K = 3 if .

Theorem 7.10. In dependence on the parameters c, t and k, the IPAD of second order of G has the form

p (7.5)

where a variant p = 3 of the nearly homocyclic abelian 3-group K of order Cl 3 K is defined as in Formula (7.4) of Theorem 7.6.

Let M = G 3 2 K be a number field with 3-class group K and first layer Cl 2 K of unramified abelian extensions.

Theorem 7.11. (Criteria for G 3 ∞ K .) Let the IPOD of K = ℚ ( d ) be of type E.8, Cl 3 K , resp. E.9, ϰ 1 K . If K with ϰ 1 K , then

・ E for 2 ≤ l 3 K ≤ 3 ,

・ l 3 K = 3 for K .

Proof. Exemplarily, we conduct the proof for H .4 , which is the most important situation for our computational applications.

Searching for the Artin pattern ϰ 1 K ∼ ( 4111 ) with G .19 and ϰ 1 K ∼ ( 2143 ) , resp. G p n K ≃ G / G ( n ) , in the descendant tree n ≥ 2 with abelian root p , unambiguously leads to the unique metabelian descendant with path G = G p ∞ K for type E.8, resp. two descendants K for type E.9. The bifurcation at the vertex p with nuclear rank two leads to a unique non-metabelian descendant with path ρ ≥ 3 for type E.8, resp. two descendants p for type E.9. The cover of l p K = ∞ is non-trivial but very simple, since it contains two elements p only. The decision whether p and L / K or p and K requires the iterated IPADs of second order p of Cl p K and p , which are listed in

The proof of Theorem 7.11, immediately justifies the following conclusions for p .

Corollary 7.3. Under the assumptions of Theorem 7.11, the second and third 3-class groups of K are given by their SmallGroups identifier [

if L , then F p 1 K for type E.8, resp. Gal ( F p 1 K / L ) for type E.9, and

if p , then G p 1 K ≃ Cl p K for type E.8, resp. p for type E.9.

In the case of a 3-class tower F p 2 K = F p 1 ( F p 1 K ) of length p ,

if K , then M : = G p 2 K : = Gal ( F p 2 K / K ) for type E.8, resp. p for type E.9, and

if K , then Cl p K for type E.8, resp. Cl p ( F p 1 K ) for type E.9.

Proposition 7.3. (Fields K ≤ L ≤ F p 1 K ≤ F p 1 L ≤ F p 2 K with IPOD of type E.8 or E.9 for p [

Proof. The results of [ [

Remark 7.5. The minimal discriminant H : = Gal ( F p 2 K / L ) of real quadratic fields p of type E.8, resp. Gal ( F p 2 K / K ) of type E.9, is indicated in boldface font adjacent to an oval surrounding the vertex, resp. batch of two vertices, which represents the associated second 3-class group L , on the branch H ′ = Gal ( F p 2 K / F p 1 L ) of the coclass tree p in

Theorem 7.12. (3-Class towers Cl p L with IPOD of type E.8 or E.9 for L / K ) Among the 3 real quadratic fields p with IPOD of type E.8 in Proposition 7.3,

・ the 2 fields ( H / H ′ ) with discriminants

H = Gal ( F p 2 K / L )

have the unique 3-class tower group p and 3-tower length p ,

・ the single field ( Gal ( F p 2 K / K ) ) with discriminant

p

has the unique 3-class tower group τ 1 K and 3-tower length K .

Among the 11 real quadratic fields Cl p L with IPOD of type E.9 in Proposition 7.3,

・ the 7 fields ( τ 1 K ) with discriminants

H

have 3-class tower group G p 2 K or p and 3-tower length p ,

・ the 4 fields ( p ) with discriminants

M = G p 2 K

have 3-class tower group K or p and 3-tower length G : = G p ∞ K : = Gal ( F p ∞ K / K ) .

Proof. Since all these real quadratic fields p have 3-capitulation

type F p ∞ K or K and p IPAD τ ( 1 ) K = [ 1 2 ; 3 2 , ( 21 ) 3 ] , and the 5

fields with p have 2^{nd} IPAD

K

whereas the 9 fields with G p ∞ K

have τ ( 2 ) K IPAD

p

the claim is a consequence of Theorem 7.9. □

Remark 7.6. The 3-principalization type E.9 of the field with l could not be computed with MAGMA versions up to V2.21-7. Finally, we succeeded to figure it out by means of V2.21-8 [

τ ∗ ( 2 ) G = [ τ 0 G ; [ τ 0 H ; τ 1 H ; τ 2 H ] H ∈ Lyr 1 G ] ,

for sporadic 3-groups G of type H.4 up to order Cl p K , characterized by the logarithmic order, ( p u , p v ) , and the SmallGroup identifier, u ≥ v ≥ 1 [

The groups in

The tree is infinite, according to Bartholdi, Bush [

For p and ϱ = 1 , we can only give the conjectural location of G.

Proposition 7.4 (Fields of type H.4 up to p [

In the range Cl p K of fundamental discriminants l = ∞ of real quadratic fields p , there exist precisely 27 cases with 3-principalization type

H.4, ϱ ≥ 3 , and IPAD τ ( 1 ) K = [ 1 2 ; ( 1 3 ) 3 ,21 ] . They share the common

second 3-class group M = G p 2 K .

Proof. The results of [ [

Remark 7.7. To discourage any misinterpretation, we point out that there are four other real quadratic fields τ 1 K with discriminants p in the range K which possess the same 3-principalization type H.4. However their second 3-class group ϰ 1 K is isomorphic to either k e r T K , L or p of order 3^{8}, which is not a sporadic group but is located on the coclass tree T K , L : Cl p K → Cl p L , and has a different IPAD τ ( 1 ) K = [ 1 2 ; 32,1 3 , ( 21 ) 2 ] . The 3-class towers of these fields are determined in [

Theorem 7.13. (3-Class towers of type H.4 up to L )

Among the 27 real quadratic fields p with type H.4 in Proposition 7.4,

・ the 11 fields ( K ) with discriminants

p

have the unique 3-class tower group p and 3-tower length ϰ 1 K ,

・ the 8 fields ( K ) with discriminants

Cl p K

have 3-class tower group ( p , p ) or l p K and 3-tower length p = 3 ,

・ the 5 fields ( Cl 3 K ) with discriminants

X .n

have the unique 3-class tower group X and 3-tower length { A,D,E,F,G,H,a,b,c,d } ,

・ the 3 fields ( n ) with discriminants

{ 1, ⋯ ,25 }

have a 3-class tower group of order at least 3^{8} and 3-tower length p = 3 .

Note that K , Cl 3 K ≃ ( 3,3 ) , X .n , and X ≠ A .

Proof. Extensions of absolute degrees 6 and 18 were constructed in steps with MAGMA [

Proposition 7.5. (Fields of type H.4 down to d < 0 [

In the range 2 ≤ l 3 K ≤ 3 of fundamental discriminants X .n ≠ A .1 of imaginary quadratic fields K = ℚ ( d ) , there exist precisely 6 cases with 3-princi- palization type H.4, d > 0 , and IPAD τ ( 1 ) K = [ 1 2 ; ( 1 3 ) 3 ,21 ] . They share the common second 3-class group A .1 .

Proof. In the table of suitable base fields [ [

which discourages an IPAD τ ( 1 ) K = [ 1 2 ; ( 1 3 ) 3 ,21 ] . □

Remark 7.8. The imaginary quadratic field with discriminant l ≥ 3 possesses the same 3-principalization type H.4, but its second 3-class group l = ∞ is isomorphic to either Cl 3 K ≃ ( 3,3 ) or p of order 3^{8}, and has the different IPAD τ ( 1 ) K = [ 1 2 ; 32 , 1 3 , ( 21 ) 2 ] . Results for this field will be given in [

Theorem 7.14. (3-Class towers of type H.4 down to p )

Among the 6 imaginary quadratic fields G / G ′ with type H.4 in Proposition 7.5,

・ the 3 fields ( G ′ ) with discriminants

( G : G ′ ) = p v

have the unique 3-class tower group v ≥ 0 and 3-tower length 0 ≤ n ≤ v ,

・ the 3 fields ( Lyr n G : = { G ′ ≤ H ≤ G | ( G : H ) = p n } ) with discriminants

G

have a 3-class tower group of order at least 3^{11} and 3-tower length G ′ .

Proof. Using the technique of Fieker [

τ ∗ ( 2 ) G = [ τ 0 G ; [ τ 0 H ; τ 1 H ; τ 2 H ] H ∈ Lyr 1 G ] ,

for sporadic 3-groups H / H ′ of type G.19 up to order T ˜ G , H : G / G ′ → H / H ′ , characterized by the logarithmic order, τ ( G ) : = τ 0 G ; ⋯ ; τ v G , and the SmallGroup identifier, G [

0 ≤ n ≤ v , since this group was called the non-CF group ϰ ( G ) : = ϰ 0 G ; ⋯ ; ϰ v G by Ascione [

G , ϰ n G : = ( k e r T ˜ G , H ) H ∈ Lyr n G , and further

0 ≤ n ≤ v , AP ( G ) : = ( τ ( G ) , ϰ ( G ) ) , and

G , τ ( 1 ) G : = τ 0 G ; τ 1 G , ϰ ( 1 ) G : = ϰ 0 G ; ϰ 1 G .

The groups in

The subtrees τ ( 2 ) G : = τ 0 G ; ( τ ( 1 ) H ) H ∈ Lyr 1 G are finite and drawn completely for 2 nd , but they are omitted in the complicated cases G , where they reach beyond order AP c ( G ) .

For Lyr 0 G = { G } , G and T G , G : G → G / G ′ , we can only give the con- jectural location of k e r ( T G , G ) = G ′ .

Proposition 7.6 (Fields of type G.19 up to ϰ 0 G [

In the range ϰ ( 1 ) G of fundamental discriminants ϰ 1 G of real quadratic fields ( G : G ′ ) = p 2 , there exist precisely ϰ ( G ) cases with 3-principalization type G.19, ϰ 1 G , consisting of two disjoint 2-cycles. Their IPAD is uniformly given by τ ( 1 ) K = [ 1 2 ; ( 21 ) 4 ] , in this range.

Proof. The results of [ [

Theorem 7.15. (3-Class towers of type G.19 up to G ′ )

The 11 real quadratic fields T G , G ′ : G → G ′ / G ″ in Proposition 7.6 with dis- criminants

k e r ( T G , G ′ ) = G

have the unique 3-class tower group G and 3-tower length c .

Proof. Extensions of absolute degrees 6 and 18 were constructed with MAGMA [

iterated IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) , ( 21 ; 1 4 , ( 2 1 ) 3 ) 3 ] was

used for the identification of n ≥ 2 , according to

Since real quadratic fields of type G.19 seemed to have a very rigid behaviour with respect to their 3-class field tower, admitting no variation at all, we were curious about the continuation of these discriminants beyond the range A ( 3, n ) . Fortunately, M. R. Bush granted access to his extended numerical results for 3 n [

Proposition 7.7. (Fields of type G.19 up to n = 2 q + r [

Proof. Private communication by M. R. Bush [

Theorem 7.16. (3-Class towers of type G.19 up to G )

Among the 64 real quadratic fields G / G ′ with type G.19 in Proposition 7.7,

・ the 11 fields with discriminants M = G / G ″ in Theorem 7.15 and the 44 fields with discriminants

c = cl ( M ) ≥ 2

(that is, together 55 fields or 86%) have τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) , ( 21 ; 1 4 , ( 2 1 ) 3 ) 3 ] ,

the unique 3-class tower group r = cc ( M ) ≥ 1 , and 3-tower length M ,

・ the 3 fields ( τ ( 1 ) G = τ 0 G ; τ 1 G ) with discriminants

G

have IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 3 1 2 ) 3 ) , ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 3 ] , the

unique 3-tower group τ 1 G of order c , and 3-tower length k ,

・ the 6 fields ( r ) with discriminants

r ≥ 3

have iterated IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 4 ] , a 3-class tower

group of order at least 1 ≤ r ≤ 2 , and 3-tower length ( T 3 , T 4 ) = ( ( A ( 3, r + 1 ) 2 ) if r = 2, M ∈ T 2 〈 243,8 〉 or r = 1 , ( 1 3 , A ( 3, r + 1 ) ) if r = 2, M ∈ T 2 〈 243,6 〉 , ( ( 1 3 ) 2 ) if r = 2, M ∈ T 2 〈 243,3 〉 or r ≥ 3. .

Proof. Similar to the proof of Theorem 7.15, using

Proposition 7.8. (Fields of type G.19 down to τ 1 G = ( ( 1 ) 4 ) for M ≃ 〈 9,2 〉 , c = 1 , r = 1, τ 1 G = ( 1 2 , ( 2 ) 3 ) for M ≃ 〈 27,4 〉 , c = 2 , r = 1, τ 1 G = ( 1 3 , ( 1 2 ) 3 ) for M ≃ 〈 81,7 〉 , c = 3 , r = 1, τ 1 G = ( ( 1 3 ) 3 ,21 ) for M ≃ 〈 243,4 〉 , c = 3 , r = 2, τ 1 G = ( 1 3 , ( 21 ) 3 ) for M ≃ 〈 243,5 〉 , c = 3 , r = 2, τ 1 G = ( ( 1 3 ) 2 , ( 21 ) 2 ) for M ≃ 〈 243,7 〉 , c = 3 , r = 2 , τ 1 G = ( ( 21 ) 4 ) for M ≃ 〈 243,9 〉 , c = 3 , r = 2, τ 1 G = ( ( 1 3 ) 3 ,21 ) for M ≃ 〈 729,44 ⋯ 47 〉 , c = 4 , r = 2, τ 1 G = ( ( 21 ) 4 ) for M ≃ 〈 729,56 ⋯ 57 〉 , c = 4 , r = 2. [

In the range τ 1 G of fundamental discriminants τ 2 G of imaginary quadratic fields c = cl ( M ) , there exist precisely 46 cases with 3-princi- palization type G.19, c , consisting of two disjoint 2-cycles, and with IPAD τ ( 1 ) K = [ 1 2 ; ( 21 ) 4 ] .

Proof. The results of [ [

Theorem 7.17. (3-Class towers of type G.19 down to 〈 81,7 〉 ≃ Syl 3 ( A 9 ) )

Among the 46 imaginary quadratic fields 〈 243, n 〉 with type G.19 in Proposition 7.8,

・ the 30 fields (65%) with discriminants

n ∈ { 4,5,7,9 }

have iterated IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 1 2 ) 3 ) 4 ] . Conjecturally,

most of them have 3-class tower group n ∈ { 44, ⋯ ,47,56,57 } of order 〈 27,3 〉 , and 3-tower length 〈 81,8 ⋯ 10 〉 , but 〈 243, n 〉 and n ∈ { 3,6,8 } cannot be excluded.

・ The 7 fields (15%) with discriminants

〈 729, n 〉

have iterated IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 2 1 ) 3 ) 4 ] , a 3-class

tower group of order at least τ 2 G , and 3-tower length G ′ ,

・ the 7 fields (15%) with discriminants

τ 2 G

have iterated IPAD of second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 3 1 2 ) 3 ) 4 ] , a proven

3-tower group p of order M = G p 2 K , and 3-tower length p = 3 ,

・ the unique field with discriminant τ 0 = ( 1 2 ) has iterated IPAD of second

order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 2 3 ) 3 ) 2 , ( 21 ; 1 4 , ( 3 2 1 ) 3 ) 2 ] , unknown 3-tower group

and 3-tower length ( τ 1 ( i ) ) 1 ≤ i ≤ 4 ,

・ the unique field with discriminant Cnt p 2 ( τ 0 , τ 1 ) has iterated IPAD of

second order τ ( 2 ) K = [ 1 2 ; ( 21 ; 1 4 , ( 3 2 1 ) 3 ) 4 ] , but unknown 3-tower group and

3-tower length M .

Proof. Similar to the proof of Theorem 7.15, using

□

In the final section §7 of [

p = 3

share the common accumulated (unordered) IPAD

τ ( 1 ) K = [ τ 0 K ; τ 1 K ] = [ 1 3 ; 32 2 1 ; ( 21 4 ) 5 , ( 2 2 1 2 ) 7 ] .

To complete the proof we had to use information on the occupation numbers of the accumulated (unordered) IPODs,

ϰ 1 K = [ 1,2,6, ( 8 ) 6 ,9, ( 10 ) 2 ,13 ] with maximal occupation number 6 for c : = cl ( M ) ,

ϰ 1 K = [ 1,2, ( 3 ) 2 , ( 4 ) 2 ,6, ( 7 ) 2 ,8, ( 9 ) 2 ,12 ] with maximal occupation number 2 for p ,

ϰ 1 K = [ ( 2 ) 2 ,5,6,7, ( 9 ) 2 , ( 10 ) 3 , ( 12 ) 3 ] with maximal occupation number 3 for τ 1 M = ( H / H ′ ) H ∈ Lyr 1 M ≃ τ 1 .

Meanwhile we succeeded in computing the second layer of the transfer target type, G 3 2 K , for the three critical fields with the aid of the computational algebra system MAGMA [

τ 2 K = [ 32 5 1 2 ; 4321 5 ; 2 5 1 3 , ( 3 2 21 5 ) 2 ; 2 4 1 4 ,32 2 1 5 ; ( 2 2 1 7 ) 3 , ( 2 3 1 5 ) 3 ] for

p ,

τ 2 K = [ 3 2 2 2 1 4 ; ( 3 2 21 5 ) 3 ; 32 2 1 5 ; ( 2 3 1 5 ) 8 ] for p , and

τ 2 K = [ 32 2 1 6 , ( 3 2 21 5 ) 3 ; 2 4 1 4 ,32 2 1 5 ; 2 2 1 7 , ( 2 3 1 5 ) 6 ] for N 1 , ⋯ , N ν .

These results admit incredibly powerful conclusions, which bring us closer to the ultimate goal to determine the precise isomorphism type of s . Firstly, they clearly show that the second 3-class groups of the three critical fields are pairwise non-isomorphic without using the IPODs. Secondly, the component with the biggest order establishes an impressively sharpened estimate for the order of G from below. The background is explained by the following lemma.

Lemma 8.1. Let G − # s ; i be a finite p-group with abelianization 1 ≤ s ≤ ν of type 1 ≤ i ≤ N s and denote by ν = 0 the logarithmic order of p with respect to the prime number p. Then the abelianizations T ( R ) of subgroups p in various layers of G admit lower bounds for G :

1) R .

2) 〈 9,2 〉 .

3) 〈 243,6 〉 , and in particular we have an equation 〈 243,8 〉 if 〈 243,4 〉 is metabelian.

Proof. The Lagrange formula for the order of 〈 243,9 〉 in terms of the index of a subgroup G reads

T G , H

and taking the p-logarithm yields

p

In particular, we have l o g p ( ( G : H ) ) = l o g p ( p n ) = n for G / G ′ ≃ ( p , p ) , 0 ≤ n ≤ 3 , and again by the Lagrange formula

p

respectively

p

with equality if and only if G 3 2 K , that is, G 3 ∞ K is abelian.

Finally, G → H is metabelian if and only if H is abelian. □

Let us first draw weak conclusions from the first layer of the TTT, i.e. the IPAD, with the aid of Lemma 8.1.

Theorem 8.1. (Coarse estimate [

The order of G / γ c ( G ) for the three critical fields K is bounded from below by c = cl ( G ) . If the maximal subgroup G with the biggest order of | G | = 3 | H | is abelian, i.e. γ c ( G ) , then the precise logarithmic order of | G | = 9 | H | is given by γ c ( G ) .

Proof. The three critical fields with discriminants

• share the common accumulated IPAD

τ ( 1 ) K = [ τ 0 K ; τ 1 K ] = [ 1 3 ; ( 32 2 1 ; ( 21 4 ) 5 , ( 2 2 1 2 ) 7 ) ] .

Consequently, Lemma 8.1 yields a uniform lower bound for each of the three fields:

k ( M ) = 0

The assumption that a maximal subgroup • having not the biggest order of M were abelian (with k ( M ) = 1 ) immediately yields the con- tradiction that

•

□

It is illuminating that much stronger estimates and conclusions are possible by applying Lemma 8.1 to the second layer of the TTT.

Theorem 8.2. (Finer estimates)

None of the maximal subgroups of M for the three critical fields K can be abelian.

The logarithmic order of k ( M ) = 0 is bounded from below by

• for M ,

k ( M ) = 1 for □ ,

G for □ .

Proof. As mentioned earlier already, computations with MAGMA [

τ 2 K = [ 32 5 1 2 ; 4321 5 ; 2 5 1 3 , ( 3 2 21 5 ) 2 ; 2 4 1 4 ,32 2 1 5 ; ( 2 2 1 7 ) 3 , ( 2 3 1 5 ) 3 ] for

d 2 ( G ) ≤ 3 ,

τ 2 K = [ 3 2 2 2 1 4 ; ( 3 2 21 5 ) 3 ; 32 2 1 5 ; ( 2 3 1 5 ) 8 ] for G , and

τ 2 K = [ 32 2 1 6 , ( 3 2 21 5 ) 3 ; 2 4 1 4 ,32 2 1 5 ; 2 2 1 7 , ( 2 3 1 5 ) 6 ] for n ∗ .

Consequently the maximal logarithmic order n is

ϰ 1 for τ 1 ( 1 ) ,

M = lo 3 ( 3 2 2 2 1 4 ) = 2 × 3 + 2 × 2 + 4 × 1 = 14 for G ,

k ( G ) = 0 for B ( j ) ≃ B ( j + 2 ) .

According to Lemma 8.1, we have j ≥ 4 .

Finally, if one of the maximal subgroups of j ≥ 7 were abelian, then Theorem 8.1 would give the contradiction that B ( 4 ) . □

Unfortunately, it was impossible for any of the three critical fields K to compute the third layer of the TTT, B ( 7 ) , that is the structure of the 3-class group of the Hilbert G -class field p of K, which is of absolute degree 54. This would have given the precise order of the metabelian group V ∈ G , according to Lemma 8.1, since p .

We also investigated whether the complete iterated IPAD of second order, M = G p 2 K , is able to improve the lower bounds in Theorem 8.2 further. It turned out that, firstly none of the additional non-normal components of p seems to have bigger order than the normal components of G = G p ∞ K , and secondly, due to the huge 3-ranks of the involved groups, the number of required class group computations enters astronomic regions.

To give an impression, we show the results for five of the 13 maximal subgroups in the case of K :

τ ( 1 ) H 1 = [ 2 2 1 2 ; 32 5 1 2 ; ( 2 3 1 5 ) 3 ; ( 3 2 21 2 ) 3 ; ( 321 4 ) 9 , ( 32 2 1 2 ) 24 ] , with 40 components,

τ ( 1 ) H 2 = [ 21 4 ; 32 5 1 2 ; 2 5 1 3 ; 2 4 1 4 ; 2 2 1 7 ; ( 31 6 ) 3 , ( 321 4 ) 33 ; ( 321 2 ) 81 ] , with 121

components,

τ ( 1 ) H 3 = [ 2 2 1 2 ; 32 5 1 2 ; 32 2 1 5 ; ( 2 2 1 7 ) 2 ; ( 321 5 ) 3 , ( 32 2 1 3 ) 6 , ( 3 2 21 2 ) 3 , ( 32 2 1 2 ) 24 ] , with

40 comp.,

τ ( 1 ) H 4 = [ 32 2 1 ; 32 5 1 2 ; 4321 5 ; ( 3 2 21 5 ) 2 ; ( 4321 3 ) 6 ; ( 431 4 ) 6 , ( 3 2 21 3 ) 6 , ( 4321 2 ) 9 , ( 3 3 1 2 ) 9 ] ,

40 comp.,

τ ( 1 ) H 5 = [ 2 2 1 2 ; 3 2 21 5 ; 32 2 1 5 ,2 4 1 4 ; 2 3 1 5 ; ( 321 3 ) 36 ] , with 40 components.

We gratefully acknowledge that our research is supported by the Austrian Science Fund (FWF): P 26008-N25.

Sincere thanks are given to Michael R. Bush (Washington and Lee University, Lexington, VA) for making available numerical results on IPADs of real quadratic fields p , and the distribution of discriminants MD : G → ℕ ∪ { ∞ } , V ↦ i n f { d | G p 2 K ( d ) ≃ V } over these IPADs [

We are indebted to Nigel Boston, Michael R. Bush and Farshid Hajir for kindly making available an unpublished database containing numerical results of their paper [

A succinct version of the present article has been delivered on July 09, 2015, within the frame of the 29ièmes Journées Arithmétiques at the University of Debrecen, Hungary [

Research supported by the Austrian Science Fund (FWF): P 26008-N25.

Mayer, D.C. (2017) Criteria for Three-Stage Towers of p-Class Fields. Advances in Pure Mathematics, 7, 135-179. https://doi.org/10.4236/apm.2017.72008

1) The restriction of the numerical results in Proposition 7.1 to the range AF : G → ℕ ∪ { 0 } , V ↦ # { d < B | G p 2 K ( d ) ≃ V } is in perfect accordance with our machine calculations by means of PARI/GP [

However, in the manual evaluation of this extensive data material for the ground state of the types a.1, a.2, a.3, and a.3*, a few errors crept in, which must be corrected at three locations: in the tables [ [

The absolute frequency of the ground state is actually given by

1382 instead of the incorrect 1386 for the union of types a.2 and a.3,

698 instead of the incorrect 697 for type a.3^{*},

2080 instead of the incorrect 2083 for the union of types a.2, a.3, and a.3*, and

150 instead of the incorrect 147 for type a.1.

(The three discriminants G were erroneously classified as type a.2 or a.3 instead of a.1.)

In the second table, two relative frequencies (percentages) should be updated:

G instead of V and

MD ( V ) ≠ ∞ instead of AF ( V ) ≠ 0 .

2) Incidentally, although it does not concern the section a of IPODs, the single field with discriminant AF was erroneously classified as type c.18, MD , instead of H.4, G . This has consequences at four locations: in the tables [ [

The absolute frequency of these types is actually given by

28 instead of the incorrect 29 for type c.18 (see also [

4 instead of the incorrect 3 for type H.4.

In the first two tables, the total frequencies should be updated, corres- pondingly:

207 instead of the incorrect 206 in [ [

66 instead of the incorrect 67 in [ [