Descending Central Series of

2-Stage Metabelian 3-Groups

1.B. All known imaginary quadratic examples:

In the following diagrams, we give the 2-stage metabelian 3-groups G = G(K₂|K) that occur for all imaginary quadratic fields K = Q(d^1/2) with discriminant -50000 < d < 0 and 3-class group of type (3,3). These recent computations of 2003 for 42 new cases in the range -50000 < d < -30000 [6] extend our own results for 22 fields with -30000 < d < -20000 of 1989 [5] and the 13 examples with -20000 < d < 0 of Heider and Schmithals in 1982 [2] . As a supplement, we give some singular cases with groups of exceptionally high order in the range -200000 < d < -50000. We denote by G = G₁ >= G₂ > ... > G_i > ... > G_m = 1 the descending central series of class m - 1 of G with G_i+1 = [G_i,G]. In particular, G₂ = [G,G] is the commutator subgroup G' of G.

Type D

36 of 77 fields (47 %)
14 (39 %) of Type D.5 and 22 (61 %) of Type D.10

(1) Group of second maximal class
(e = 3)
G = G^(4,5)((alpha,beta,gamma,delta),rho)
in ZEF 1a(4,5)
with (alpha,beta,gamma,delta) = (0,0,-1,1), rho = 0,
resp. (alpha,beta,gamma,delta) = (1,1,-1,1), rho = 0,
of class 3 and of order 3⁵ = 243

				K₂
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₄ = 1
				\|\|
				G₃ = (3,3)
				\|
				G₂ = (3,3,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -4027, resp.
discriminant d = -12131
Principalization type of K in N₁,...,N₄:
D.10, (1,1,2,3), resp. D.5, (1,2,1,2) [2,5] ,
both designated by D in [1]

Remarks:
(1,1,2,3) and (1,2,1,2) are the only two principalization types where the associated Galois group G = G(K₂|K) is uniquely determined.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -8751, -19651, -21224, -22711, -24904, -26139, -28031, -28759 of principalization type D.10, (1,1,2,3) [2,5] , resp.
d = -19187, -20276, -20568, -24340, -26760 of principalization type D.5, (1,2,1,2) [2,5] .

The Hilbert 3-class field tower K = K₀ <= K₁ <= K₂ <= ... of all these fields terminates after 2 steps with K₂, i. e., 3 does not divide the class number of K₂. This was shown by Scholz and Taussky in [1] and, with a different proof, by Brink and Gold in [3] .

Results discovered 2003/04/19 - 22 and 2003/05/09:
According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -31639, -31999, -32968, -34507, -35367, -41583, -41671, -43307
of principalization type D.5, (1,2,1,2) and
d = -34088, -36807, -40299, -40692, -41015, -42423, -43192, -44004, -45835, -46587, -48052, -49128, -49812
of principalization type D.10, (1,1,2,3).

Type E

18 of 77 fields (23 %)
3 (17 %) of Type E.6, 2 (11 %) of Type E.8,
9 (50 %) of Type E.9, and 4 (22 %) of Type E.14

(2) Group of second maximal class
(e = 3)
G = G^(6,7)((alpha,beta,gamma,delta),rho)
in ZEF 2a(6,7)
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0,
of class 5 and of order 3⁷ = 2187

				K₂
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₆ = 1
				\|
				G₅ = (3)
				\|
				G₄ = (3,3)
				\|\|
				G₃ = (3²,3,3)
				\|
				G₂ = (3²,3²,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -9748
Principalization type of K in N₁,...,N₄:
E.9, (1,1,3,2) or equivalently (1,2,1,3) [2,5] ,
designated by E in [1]

Remarks:
Here, the principalization type (1,1,3,2) does not determine the associated Galois group G = G(K₂|K) uniquely.
However, Scholz and Taussky [1] provide additional information for d = -9748 mentioning that a = 4 in the associated symbolic order X_a = (X^a,XY,Y²,3+3X+3X²) whence G' = G₂ = Z[X,Y]/X_a = (3²,3²,3), and this property determines G uniquely, up to the sign of gamma.

The Hilbert 3-class field tower K = K₀ <= K₁ <= K₂ <= ... of the field K = Q((-9748)^1/2) terminates after 2 steps with K₂, i. e., 3 does not divide the class number of K₂. That is the only claim by Scholz and Taussky in [1] and not a statement for all fields with associated symbolic order X_a. Heider and Schmithals erroneously remark in [2] that Scholz and Taussky proved G₄ = 1 for the fields K with associated symbolic order X_a, whereas they really have G₄ = (3,3). This led to a misinterpretation by Brink and Gold in [3] , where they construct a 3-stage metabelian 3-group that could possibly be the Galois group G(M|K) of an unramified cyclic cubic extension M of K₂ when K has the associated symbolic order X_a.
But no explicit example is known up to now for a 3-class field tower of height greater than 2.

Results discovered 2003/01/14 - 18:
According to recent computations of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,3) by Karim Belabas with the aid of PARI, exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -22395, -22443, -27640 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3) [5] ,
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0, resp.
d = -15544, -18555, -23683 of principalization type E.6, (1,1,2,2) [2,5] ,
with (alpha,beta,gamma,delta) = (1,-1,1,1), rho = 0, resp.
d = -16627 of principalization type E.14, (2,3,1,1) [2,5] ,
with (alpha,beta,gamma,delta) = (0,-1,+-1,1), rho = 0,
which are all designated by E in [1] .

Results discovered 2003/03/23 and 2003/05/24:
According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -31271, -34867, -37988, -39736, -42619, -42859, -43847, -45887, -48472, -48667.

More detailed, according to [6] , we finally have
d = -37988, -39736, -45887, -48472, -48667 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3),
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0,
d = -31271, -42859, -43847 of principalization type E.14, (2,3,1,1),
with (alpha,beta,gamma,delta) = (0,-1,+-1,1), rho = 0, and
d = -34867, -42619 of the newly discovered principalization type E.8, (1,2,3,1), the unique one with 3 fixed points (A,A,A,B) and
with (alpha,beta,gamma,delta) = (1,0,-1,1), rho = 0.

Type H

13 of 77 fields (17 %)

(3) Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (?,1,?,1), rho = 1,
of class 4 and of order 3⁶ = 729
with rho != beta - 1

				K₂
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3²,3,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -3896
Principalization type of K in N₁,...,N₄:
H.4, (2,1,1,1) [2,5] ,
designated by H in [1]

Remarks:
Again, the principalization type (2,1,1,1) does not determine the associated Galois group G = G(K₂|K) uniquely.

Results discovered 2003/01/14 and 2003/02/17:
However, according to recent computations of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,3,3) by Karim Belabas with the aid of PARI and Claus Fieker with the aid of MAGMA, at least delta = 1, the class 4, and the order 729 of G are determined uniquely.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -6583, -23428, -25447, -27355, -27991 of the same principalization type H.4, (2,1,1,1) [2,5] ,
designated by H in [1] .

Results discovered 2003/04/22 and 2003/05/09:
According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -36276, -37219, -37540, -39819, -41063,
and:
d = -43827, -46551.

Type G.19

2 of 77 fields (3 %)

(4) Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (?,0,?,1), rho = -1,
of class 4 and of order 3⁶ = 729
with rho = beta - 1 != 0

				K₂
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3,3,3,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -12067
Principalization type of K in N₁,...,N₄:
G.19, (2,1,4,3) [2,5] ,
designated by G in [1]

Remarks:
As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/01/14:
But according to the recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (3,3,3,3) by Karim Belabas with the aid of PARI, at least beta = 0, delta = 1, rho = -1, the class 4, and the order 729 of G are determined uniquely.

Result discovered 2003/05/10:
According to [6] , we now have another occurrence of this rare case:
d = -49924.

Type F

2 of 77 fields (3 %)
1 (50 %) of Type F.11, 1 (50 %) of Type F.12

(5) Group of lower than second maximal class
(e = 5)
G = G^(6,9)((alpha,beta,gamma,delta),rho)
in ZEF 1a(6,9)
with (alpha,beta,gamma,delta) = (?,?,0,0), rho = 0,
of class 5 and of order 3⁹ = 19683

				K₂
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₆ = 1
				\|\|
				G₅ = (3,3)
				\|\|
				G₄ = (3,3,3,3)
				\|\|
				G₃ = (3²,3²,3,3)
				\|
				G₂ = (3²,3²,3²,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -27156
Principalization type of K in N₁,...,N₄:
F.11, (1,3,2,1) [5] ,
designated by F in [1]

Remarks:
Again, the principalization type (1,3,2,1) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/02/13:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,9,3) by Claus Fieker with the aid of MAGMA, at least gamma = 0, delta = 0, rho = 0, the class 5, and the order 19683 of G are determined uniquely.

Results discovered 2003/03/23, 2003/05/27 - 31, and 2003/06/10:
According to [6] and computations of Karim Belabas with the aid of PARI, we now have other occurrences of this rare case:
d = -31908, -135587 of the newly discovered principalization type F.12, (2,1,3,1) ~ (3,2,1,1).

According to the supplements section [7] of [6] , we finally have other occurrences of this rare case:
d = -67480, -104627 of the newly discovered principalization type F.13, (2,1,1,3),
and d = -124363 of the newly discovered principalization type F.7, (2,1,1,2).

Results discovered 2003/09/16:
According to the supplements section [7] of [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -160403, -184132, -189959 of type F.12, (2,1,3,1) ~ (3,2,1,1)
and d = -167064 of type F.13, (2,1,1,3).

Type G.16 and a Variant of Type H

4 of 77 fields (5 %) ... Type G.16
2 of 77 fields (3 %) ... Variant of Type H.4

(6) Group of second maximal class
(e = 3)
G = G^(7,8)((alpha,beta,gamma,delta),rho)
in ZEF 2b(7,8)
with (alpha,beta,gamma,delta) = (?,0,?,1), rho = +-1,
of class 6 and of order 3⁸ = 6561

				K₂
				\|
				F₆
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|
				G₅ = (3,3)
				\|
				G₄ = (3²,3)
				\|\|
				G₃ = (3²,3²,3)
				\|
				G₂ = (3³,3²,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -17131
Principalization type of K in N₁,...,N₄:
G.16, (1,2,4,3) [2,5] ,
designated by G in [1]

Remarks:
As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Results discovered 2003/02/17:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (27,9,3) by Claus Fieker with the aid of MAGMA, at least beta = 0, delta = 1, the class 6, and the order 6561 of G are determined uniquely.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -24884, -28279 of the same principalization type G.16, (1,2,4,3) ~ (2,1,3,4) [5] ,
with (alpha,beta,gamma,delta) = (1,0,0,1), rho = 0 or (alpha,beta,gamma,delta) = (?,0,?,1), rho = +-1,
designated by G in [1] , resp.
d = -21668 of principalization type H.4.V1, (2,1,1,1) [5] ,
with (alpha,beta,gamma,delta) = (1,-1,-1,1), rho = 0 or (alpha,beta,gamma,delta) = (?,-1,?,1), rho = +-1,
designated by H in [1] .

Results discovered 2003/03/23:
According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -34027 of principalization type H.4.V1, (2,1,1,1) and d = -35539 of principalization type G.16, (1,2,4,3) ~ (2,1,3,4).

A Variant of Type G.19

(7) Group of lower than second maximal class
(e = 5)
G = G^(7,10)((alpha,beta,gamma,delta),rho)
in ZEF 1b(7,10)
with (alpha,beta,gamma,delta) = (?,?,?,0), rho = +1,
of class 6 and of order 3¹⁰ = 59049

				K₂
				\|
				F₆
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|\|
				G₅ = (3,3,3)
				\|\|
				G₄ = (3²,3,3,3)
				\|\|
				G₃ = (3²,3²,3²,3)
				\|
				G₂ = (3³,3²,3²,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal and up to |d| < 150000 unique occurrence:
discriminant d = -96827
Principalization type of K in N₁,...,N₄:
G.19, (2,1,4,3) [7] ,
designated by G in [1]

Remarks:
As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/05/29:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (27,9,9,3) by Karim Belabas with the aid of PARI, at least delta = 0, the class 6, and the order 59049 of G are determined uniquely.

Result discovered 2003/09/16:
According to the supplements section [7] of [6] and a computation of Karim Belabas with the aid of PARI, we now have a further occurrence of this case:
d = -156452.

A Variant of Type G.16 and another Variant of Type H

(8) Group of lower than second maximal class
(e = 5)
G = G^(7,10)((alpha,beta,gamma,delta),rho)
in ZEF 1b(7,10)
with (alpha,beta,gamma,delta) = (?,?,?,0), rho = -1,
of class 6 and of order 3¹⁰ = 59049

				K₂
				\|
				F₆
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|\|
				G₅ = (3,3,3)
				\|\|
				G₄ = (3²,3,3,3)
				\|\|
				G₃ = (3²,3²,3²,3)
				\|
				G₂ = (3²,3²,3²,3²)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal and up to |d| < 150000 unique occurrence:
discriminant d = -135059
Principalization type of K in N₁,...,N₄:
G.16.V, (1,2,4,3) [7] ,
designated by G in [1]

Remarks:
As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/06/10:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,9,9) by Karim Belabas with the aid of PARI, at least delta = 0, rho = -1, the class 6, and the order 59049 of G are determined uniquely.

Results discovered 2003/09/16 and 2004/01/06:
According to the supplements section [7] of [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -185747 of type G.16.V, (1,2,4,3),
and d = -186483 of type H.4.V2, (2,1,1,1).

A Variant of Type F.13

(9) Group of lower than second maximal class
(e = 5)
G = G^(8,11)((alpha,beta,gamma,delta),rho)
in ZEF 2a(8,11)
with (alpha,beta,gamma,delta) = (?,?,?,0), rho = 0,
of class 7 and of order 3¹¹ = 177147

				K₂
				\|
				F₇
				\|
				F₆
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₈ = 1
				\|
				G₇ = (3)
				\|
				G₆ = (3,3)
				\|\|
				G₅ = (3²,3,3)
				\|\|
				G₄ = (3²,3²,3,3)
				\|\|
				G₃ = (3³,3²,3²,3)
				\|
				G₂ = (3³,3³,3²,3)
	/		/		\		\
M₁		M₂				M₃		M₄
	\		\		/		/
				G₁ = G

Minimal and up to |d| < 200000 unique occurrence:
discriminant d = -159208
Principalization type of K in N₁,...,N₄:
F.13.V, (2,1,1,3) [7] ,
designated by F in [1]

Remarks:
As above, the principalization type (2,1,1,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/08/10:
However, according to an investigation of the transfers ("Verlagerungen") to the 4 maximal subgroups M₁,...,M₄ of G, at least delta = 0, rho = 0, the class 7, and the order 177147 of G are determined uniquely.

References:

[1] Arnold Scholz und Olga Taussky,
Die Hauptideale der kubischen Klassenkörper
imaginär quadratischer Zahlkörper,
J. reine angew. Math.171 (1934), 19 - 41

[2] Franz-Peter Heider und Bodo Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25

[3] James R. Brink and Robert Gold,
Class field towers of imaginary quadratic fields,
manuscripta math. 57 (1987), 425 - 450

[4] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, K�ln, 1989

[5] Daniel C. Mayer,
Principalization in complex S₃-fields,
Congressus Numerantium 80 (1991), 73 - 87

[6] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of all Quadratic Fields with Discriminant -50000 < d < 0
and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2003

[7] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of selected Quadratic Fields with Discriminant -200000 < d < -50000
and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2004

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