1. Transfers ("Verlagerungen").Our Target.If the 3-class group Syl3(C(K)) of an arbitrary base field K is of type (3,3), then the structure of the automorphism group G = Gal(K2|K) of the 2nd Hilbert 3-class field K2 of K over K as a 2-stage metabelian 3-group in ZEF determines the principalization type (k(1),...,k(4)) of K in the four unramified cyclic cubic extensions N1,...,N4 of K.
How do we approach this target? Artin's Idea. 
For this enterprise it is necessary to investigate the
transfer VG,M of the commutator factor group G/G' to the commutator factor group M/M' for each of the 4 maximal subgroups M = Mj (1 <= j <= 4) of G. Class field theory establishes the connection between the transfer VG,M : G/G' --> M/M' and the class extension homomorphism jN|K : Syl3(C(K)) --> Syl3(C(N)), since G/G' ~ Gal(K1|K) ~ Syl3(C(K)) and Mj/Mj' ~ Gal((Nj)1|Nj) ~ Syl3(C(Nj)).
PROPOSITION 1.1. Assumptions: Let m,n be rational integers, such that 2 <= m <= n and let G be a 2-stage metabelian 3-group in ZEF(m,n) with generators x,y, such that the 4 maximal subgroups Mj (1 <= j <= 4) of G are given by Mj = < g(j), G' > where g(1) = x, g(2) = y, g(3) = xy, g(4) = xy-1. Finally, denote by Vj the transfer from G/G' to Mj/Mj'. Claims: 1. The commutator subgroups are Mj' = (G')g(j)-1 (1 <= j <= 4). 2. The transfers are given by a) Vj(gG') = g3Mj' for g in G - Mj b) Vj(uG') = u1+h+h2Mj' for u in Mj with arbitrary h in G - Mj Remark: u1+h+h2 = u3[u,h]3[[u,h],h] (mod Mj'). |
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2. The Class Groups of N1,...,N4.Commutator Subgroups of the Maximal Subgroups.We can considerably refine claim 1 of the preceding Proposition 1.1. THEOREM 2.1. Assumptions: Let m be a rational integer, such that m >= 3, and let G = G(m)(a,b,c) be a 2-stage metabelian 3-group of maximal class in ZEF(m,m) with generators s,s1, where we assume s in G - C2 and s1 in C2 - G' in the case of m > 3, i. e., G' < C2 < G, such that the 4 maximal subgroups Mj (1 <= j <= 4) of G are given by Mj = < g(j), G' > where g(1) = s1, g(2) = s, g(3) = ss1, g(4) = ss1-1. Claims: 1. The commutator subgroups are Mj' = G3 and thus Syl3(C(Nj)) = Mj/Mj' = (3,3) uniformly for 2 <= j <= 4. 2. An exceptional role is played by M1' = < sm-1c >: a) M1' = 1 and Syl3(C(N1)) = M1 = A(3,m-1) (for m = 4, we can also have M1 = (3,3,3)) for c = 0, i. e., G in ZEF a(m,m) b) M1' = Gm-1 = (3) and Syl3(C(N1)) = M1/M1' = A(3,m-2) for c != 0, i. e., G in ZEF b(m,m) 3. M1 is an abelian normal subgroup of G for c = 0. 4. The Mj with j != 1 are abelian normal subgroups of G iff m = 3.
THEOREM 2.2. Assumptions: Let m,n be a rational integers, such that 4 <= m < n <=2m - 3, i. e., e = n - m + 2 >= 3, and let G = G(m,n)((a,b,c,d),r) be a 2-stage metabelian 3-group of non-maximal class in ZEF(m,n) with generators x,y, such that G3 = < x3,y3,G4 >, where we assume x in G - Cs and y in Cs - G' in the case of s < m - 1, i. e., G in ZEF 2(m,n), such that the 4 maximal subgroups Mj (1 <= j <= 4) of G are given by Mj = < g(j), G' > where g(1) = y, g(2) = x, g(3) = xy, g(4) = xy-1. Claims: Introducing some additional notation v3 = [y,x]x-1, w3 = [y,x]y-1, s4 = (y3)x-1, s5 = s4x-1, t4 = (x3)y-1, t5 = t4y-1, S4 = < s4 > * < s5 >, T4 = < t4 > * < t5 > we have: 1. M1' = < w3,T4 > and a) Syl3(C(N1)) = M1/M1' = A(3,m-1) for r = 0, i. e., G in ZEF a(m,n) (for m = 4, we can also have M1/M1' = (3,3,3)) b) Syl3(C(N1)) = M1/M1' = A(3,m-2) for r != 0, i. e., G in ZEF b(m,n) (for m = 5, we can also have M1/M1' = (3,3,3)) 2. M2' = < v3,S4 > and Syl3(C(N2)) = M2/M2' = A(3,e) (for e = 3, we can also have M2/M2' = (3,3,3)) 3. M3' = < v3w3,G4 > and Syl3(C(N3)) = M3/M3' = (9,3) or (3,3,3) 4. M4' = < v3w3-1,G4 > and Syl3(C(N4)) = M4/M4' = (9,3) or (3,3,3) 5. In particular, none of the maximal normal subgroups M1,...,M4 is abelian.
Insider's Know How. In the special case of a quadratic base field K, the structure of G = Gal(K2|K) determines also the family of 3-class numbers (h1,...,h4) of the absolute cubic subfields L1,...,L4 of the normal S3-fields N1,...,N4 between K1 and K, and, to a certain extent, also vice versa. For the proof of the following theorem we use the class number formulas of Scholz for the principal factorization types (PFT) Alpha and Delta. THEOREM 2.3. Assumptions: Let K be a quadratic number field with discriminant d and with 3-class group of type (3,3). Denote by G = Gal(K2|K) in ZEF(m,n) with 3 <= m <= n the automorphism group of the 2nd Hilbert 3-class field K2 of K over K. Claims: 1. If G is of maximal class, i. e., m = n, then K must be a real quadratic field with d > 0 and (h1,...,h4) = (3q,3,3,3), where the exponent q is given by [a) if G in ZEF 2a(m,m) with odd m, then q = (m - 1)/2 and L1 is of PFT Alpha, a case which is conjectured to be impossible and did never occur in numerical examples,] b) if G in ZEF 2a(m,m) with even m, then q = (m - 2)/2 and L1 is of PFT Delta, c) if G in ZEF 2b(m,m) with even m, then q = (m - 2)/2 and L1 is of PFT Alpha, [d) if G in ZEF 2b(m,m) with odd m, then q = (m - 3)/2 and L1 is of PFT Delta, but this case is definitely impossible, since G in ZEF 2b(m,m) implies total principalization in L1.] L2,L3,L4 are always of PFT Alpha. 2. If G is not of maximal class, i. e., 4 <= m < n, resp. e = n - m + 2 >= 3, and if K is a complex quadratic field with d < 0, then L1,...,L4 are of PFT Alpha, e must be odd, G is either in ZEF a(m,n) with even m or in ZEF b(m,n) with odd m, and (h1,...,h4) = (3q(1),3q(2),3,3), where the exponents are given by a) if G in ZEF a(m,n) with even m, then q(1) = (m - 2)/2 b) if G in ZEF b(m,n) with odd m, then q(1) = (m - 3)/2 c) q(2) = (e - 1)/2 3. If K is a complex quadratic field with d < 0, then G is not of maximal class, i. e., 4 <= m < n, resp. e = n - m + 2 >= 3, e = 2*q(2) + 1 must be odd, G is either in ZEF a(m,n) with even m = 2*q(1) + 2 or in ZEF b(m,n) with odd m = 2*q(1) + 3, and we have the following particular implications: a) if (h1,...,h4) = (3,3,3,3), i. e., q(1) = q(2) = 1, then G in ZEF 1a(4,5) or ZEF 1b(5,6) b) if (h1,...,h4) = (9,3,3,3), i. e., q(1) = 2, q(2) = 1, then G in ZEF 2a(6,7) or ZEF 2b(7,8) c) if (h1,...,h4) = (9,9,3,3), i. e., q(1) = q(2) = 2, then G in ZEF 1a(6,9) or ZEF 1b(7,10) d) if (h1,...,h4) = (27,9,3,3), i. e., q(1) = 3, q(2) = 2, then G in ZEF 2a(8,11) or ZEF 2b(9,12) |
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3. Principalization Types.Explicit terms for the images under the transfers.PROPOSITION 3.1. Assumptions: Let m >= 3 be a rational integer and G = G(m)(a,b,c) in ZEF (m,m) a 2-stage metabelian 3-group of maximal class with generators s,s1, where we assume s in G - C2 and s1 in C2 - G' in the case of m > 3, i. e., G' < C2 < G. Finally, let g be an element of G with representation g = sis1k (mod G'), where -1 <= i,k <= 1, and denote by Vj the transfer from G/G' to Mj/Mj' (1 <= j <= 4). Claims: 1. V1(g*G') = sm-1ai+bk*< sm-1c > 2. a) Vj(g*G') = s2ai+bk*1 for 2 <= j <= 4, if m = 3 b) Vj(g*G') = 1*G3 for 2 <= j <= 4, if m >= 4 The transfer kernels. THEOREM 3.2. Assumptions: similar as in Proposition 3.1. Claims: 1. Cases having the total preimage group G/G' as kernel: a) c != 0 ==> Ker V1 = G/G' b) a = b = 0 ==> Ker V1 = G/G' c) a = b = 0, m = 3 ==> Ker Vj = G/G' for 1 <= j <= 4 d) m >= 4 ==> Ker Vj = G/G' for 2 <= j <= 4 2. Cases having a non-trivial subgroup of G/G' as kernel: a) (a,b,c) = (1,0,0) ==> Ker V1 = M1/G' b) (a,b,c) = (1,0,0), m = 3 ==> Ker Vj = M1/G' for 1 <= j <= 4 c) (a,b,c) = (0,+-1,0) ==> Ker V1 = M2/G' 3. The Principalization Types for Groups of Maximal Class are: a) (a,b,c) = (1,0,0), m = 3 ==> (k(1),...,k(4)) = (1,1,1,1) b) (a,b,c) = (1,0,0), m >= 4 ==> (k(1),...,k(4)) = (1,0,0,0) c) (a,b,c) = (0,+-1,0), m >= 4 ==> (k(1),...,k(4)) = (0,1,0,0) d) (a,b,c) = (0,0,0) ==> (k(1),...,k(4)) = (0,0,0,0) e) c != 0 ==> (k(1),...,k(4)) = (0,0,0,0) |
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