Research Frontier 2013

2-Class Field Towers of Type (2,2)

having a Single Stage or Two Stages



Karl-Franzens University Graz, left side

Statistics and Minima:

  • Here we present the results of an application
    of our New Algorithm for determining the
    Second p-Class Group Gp2(K)
    of a number field K
    by computing the Kernels and Targets
    of its Transfers to maximal subgroups.

    We set p = 2 and obtain the complete
    statistical evaluation of the first 100
    complex or real quadratic fields
    K = Q(D1/2)
    or bicyclic biquadratic Dirichlet fields
    K = Q((-1)1/2,D1/2)
    having a 2-class group of type (2,2).
    The distribution visualizes
    the population by G22(K)
    of vertices on the
    coclass graph G(2,1) of
    2-groups of coclass 1,
    consisting of

  • the abelian group of type (2,2),
  • the quaternion group Q(8),
  • the generalized quaternion groups Q(2n),
  • the dihedral groups D(2n),
  • the semidihedral groups S(2n).


Karl-Franzens University Graz, centre with 8 figures

  • Top-Down Algorithm:

    *************************************************************************

    First Step: with the aid of PARI/GP we compute a list of
    the first 100 complex, resp. real, quadratic number fields K,
    resp. bicyclic biquadratic special Dirichlet fields K,
    having a 2-class group Cl2(K) of type (2,2).
    --------------------------------------------------------------

    Second Step: by means of MAGMA we iterate through the list
    of discriminants d(K) = D computed in the first step and determine
    1. the 2-principalization kernel of K in its three
      unramified cyclic quadratic extensions N1,N2,N3
      (the transfer kernel type TKT),
    2. the structure of the 2-class groups of N1,N2,N3
      (the transfer target type TTT).
    --------------------------------------------------------------

    Third Step: we use our unambiguous criteria
    to identify the second 2-class group G22(K) of K
    by the combination of TKT and TTT in the following table.
    --------------------------------------------------------------

    G22(K) TKT Cl2(N1) Cl2(N2) Cl2(N3) Cl2(F21(K))
    <4,2> ~ (2,2) (0,0,0) (2) (2) (2) 1
    <8,4> ~ Q(8) (1,2,3) (4) (4) (4) (2)
    <8,3> ~ D(8) (0,3,2) (4) (2,2) (2,2) (2)
    <16,9> ~ Q(16) (1,3,2) (8) (2,2) (2,2) (4)
    <16,8> ~ S(16) (2,3,2) (8) (2,2) (2,2) (4)
    <16,7> ~ D(16) ↑ (0,3,2) (8) (2,2) (2,2) (4)
    <32,20> ~ Q(32) ↑ (1,3,2) (16) (2,2) (2,2) (8)
    <32,19> ~ S(32) ↑ (2,3,2) (16) (2,2) (2,2) (8)
    <32,18> ~ D(32) ↑2 (0,3,2) (16) (2,2) (2,2) (8)
    <64,54> ~ Q(64) ↑2 (1,3,2) (32) (2,2) (2,2) (16)


    Remarks:
    1. Only the groups <4,2> and <8,4> can be identified
    by their unique TKT or TTT alone.
    The group <8,3> is characterized by its TTT.

    2. Neither the coarse TKTs by Kisilevsky for complex quadratic fields,
    or by Benjamin and Snyder for real quadratic fields K,
    nor the quadratic residue conditions for prime divisors
    of the discriminant by Azizi, Mouhib and Taous
    for bicyclic biquadratic Dirichlet fields K
    are able to determine the order of G22(K)
    for higher excited states of D(2n), Q(2n) and S(2n).

    3. Our new criteria use the TTT to resolve the
    infinite sequences sharing a common TKT.
    Of course, they would have been useless
    in times before availability of Pari/GP and Magma.

    *************************************************************************

  • Statistics:

    G22(K) Abelian Dihedral Quaternion Quaternion Semidihedral
    (2,2) D(2n), n ≥ 3 Q(8) Q(2n), n ≥ 4 S(2n), n ≥ 4
    K Dirichlet 39 20 + 8 ↑ + 5 ↑2 9 10 + 8 ↑ + 1 ↑2 0
    K complex 38 8 + 1 ↑ 18 18 + 10 ↑ + 1 ↑2 4 + 2 ↑
    K real 59 24 10 5 + 1 ↑ 1


    The 2-tower consists of a single stage
    if and only if G22(K) is abelian.

    For real quadratic fields, single stage 2-towers are dominating with 59%
    and excited states occur with considerable delay.

    For complex quadratic fields, all quaternion groups together
    are populated most densely with 47%.
    However, if different variants of quaternion groups are separated,
    then abelian groups with 38% are the high-champs again.

    For bicyclic biquadratic Dirichlet fields,
    single stage 2-towers are slightly dominating with 39%
    closely followed by dihedral (33%) and quaternion (28%) groups.

    It was proved in several steps by Abdelmalek Azizi
    [Az] , [AAI] , [Az1]
    and by Ali Mouhib in cooperation with Azizi
    that semidihedral groups are impossible
    for bicyclic biquadratic Dirichlet fields.


  • Minimal Discriminants:

    G22(K) Abelian Dihedral Quaternion Quaternion Semidihedral
    (2,2) D(2n), n ≥ 3 Q(8) Q(2n), n ≥ 4 S(2n), n ≥ 4
    |D|min of K Dirichlet 51 123 120 168 impossible
    771 744
    1563 3048
    |D|min of K complex 84 408 120 195 340
    1515 555 2132
    2355
    Dmin of K real 680 1160 520 1740 3965
    4520


    Here we list minimal discriminants of ground states in the first row,
    and of excited states in additional rows.

    *************************************************************************

  • Details for 100 discriminants
    of bicyclic biquadratic Dirichlet fields
    K = Q((-1)1/2,D1/2)
    with complex quadratic subfields
    k = Q((-D)1/2):

    Cases with non-abelian group G22(K)
    are printed in red or green boldface font.

    Cases where the complex quadratic subfield k is also of type (2,2)
    are marked by ≅ if the TKTs of K and k coincide,
    by # if k has an abelian group G22(k),
    and by ⊗ if k has a semidihedral group G22(k).

    No. D factors TKT κ(K) G22(K)
    1 51 3*17 a.1 (000) (2,2)
    2 119 7*17 a.1 (000) (2,2)
    3 120 2*3*5 ≅Q.5 (123) Q(8)
    4 123 3*41 d.8 (032) D(8)
    5 168 2*3*7 #Q.6 (132) Q(16)
    6 187 11*17 a.1 (000) (2,2)
    7 267 3*89 a.1 (000) (2,2)
    8 280 2*5*7 ≅Q.6 (132) Q(16)
    9 287 7*41 d.8 (032) D(8)
    10 312 2*3*13 ≅Q.6 (132) Q(16)
    11 339 3*113 d.8 (032) D(8)
    12 340 (2*)5*17 ⊗d.8 (032) D(8)
    13 391 17*23 a.1 (000) (2,2)
    14 411 3*137 d.8 (032) D(8)
    15 440 2*5*11 ≅Q.6 (132) Q(16)
    16 451 11*41 d.8 (032) D(8)
    17 511 7*73 a.1 (000) (2,2)
    18 527 17*31 a.1 (000) (2,2)
    19 623 7*89 a.1 (000) (2,2)
    20 679 7*97 a.1 (000) (2,2)
    21 696 2*3*29 ≅Q.5 (123) Q(8)
    22 699 3*233 a.1 (000) (2,2)
    23 728 2*7*13 ≅Q.6 (132) Q(16)
    24 744 2*3*31 #Q.6↑ (132) Q(32)
    25 760 2*5*19 ≅Q.6↑ (132) Q(32)
    26 771 3*257 d.8↑ (032) D(8)
    27 779 19*41 d.8 (032) D(8)
    28 803 11*73 a.1 (000) (2,2)
    29 843 3*281 a.1 (000) (2,2)
    30 888 2*3*37 ≅Q.6↑ (132) Q(32)
    31 920 2*5*23 ≅Q.6 (132) Q(16)
    32 1059 3*353 d.8↑ (032) D(8)
    33 1064 2*7*19 #Q.6 (132) Q(16)
    34 1144 2*11*13 ≅Q.5 (123) Q(8)
    35 1203 3*401 a.1 (000) (2,2)
    36 1207 17*71 a.1 (000) (2,2)
    37 1272 2*3*53 ≅Q.5 (123) Q(8)
    38 1343 17*79 a.1 (000) (2,2)
    39 1347 3*449 a.1 (000) (2,2)
    40 1460 (2*)5*73 ⊗d.8 (032) D(8)
    41 1464 2*3*61 ≅Q.6↑ (132) Q(32)
    42 1563 3*521 d.8↑2 (032) D(8)
    43 1687 7*241 a.1 (000) (2,2)
    44 1691 19*89 a.1 (000) (2,2)
    45 1707 3*521 d.8↑2 (032) D(8)
    46 1720 2*5*43 ≅Q.5 (123) Q(8)
    47 1779 3*593 d.8 (032) D(8)
    48 1799 7*257 d.8↑ (032) D(8)
    49 1819 17*107 a.1 (000) (2,2)
    50 1843 19*97 a.1 (000) (2,2)
    51 1851 3*617 a.1 (000) (2,2)
    52 1880 2*5*47 ≅Q.6↑ (132) Q(32)
    53 1896 2*3*79 #Q.6↑ (132) Q(32)
    54 1923 3*641 a.1 (000) (2,2)
    55 1927 41*47 d.8 (032) D(8)
    56 1940 (2*)5*97 ⊗d.8 (032) D(8)
    57 1972 (2*)17*29 ⊗d.8 (032) D(8)
    58 1976 2*13*19 ≅Q.5 (123) Q(8)
    59 2024 2*11*23 #Q.6 (132) Q(16)
    60 2047 23*89 a.1 (000) (2,2)
    61 2123 11*193 a.1 (000) (2,2)
    62 2132 (2*)13*41 ⊗d.8↑ (032) D(8)
    63 2147 19*113 d.8 (032) D(8)
    64 2191 7*313 d.8 (032) D(8)
    65 2227 17*131 a.1 (000) (2,2)
    66 2231 23*97 a.1 (000) (2,2)
    67 2260 (2*)5*113 ⊗d.8↑ (032) D(8)
    68 2263 31*73 a.1 (000) (2,2)
    69 2283 3*761 d.8 (032) D(8)
    70 2360 2*5*59 ≅Q.6↑ (132) Q(32)
    71 2363 17*139 a.1 (000) (2,2)
    72 2424 2*3*101 ≅Q.5 (123) Q(8)
    73 2427 3*809 d.8↑2 (032) D(8)
    74 2471 7*353 d.8↑ (032) D(8)
    75 2472 2*3*103 #Q.6 (132) Q(16)
    76 2516 (2*)17*37 ⊗d.8 (032) D(8)
    77 2552 2*11*29 ≅Q.5 (123) Q(8)
    78 2563 11*233 a.1 (000) (2,2)
    79 2571 3*857 d.8↑2 (032) D(8)
    80 2599 23*113 d.8 (032) D(8)
    81 2616 2*3*109 ≅Q.6 (132) Q(16)
    82 2643 3*881 d.8 (032) D(8)
    83 2651 11*241 a.1 (000) (2,2)
    84 2680 2*5*67 ≅Q.5 (123) Q(8)
    85 2728 2*11*31 #Q.6↑ (132) Q(32)
    86 2740 (2*)5*137 ⊗d.8↑ (032) D(8)
    87 2747 41*67 d.8 (032) D(8)
    88 2759 31*89 a.1 (000) (2,2)
    89 2771 17*163 a.1 (000) (2,2)
    90 2787 3*929 a.1 (000) (2,2)
    91 2839 17*167 a.1 (000) (2,2)
    92 2859 3*953 d.8↑2 (032) D(8)
    93 2863 7*409 d.8↑ (032) D(8)
    94 2911 41*71 d.8 (032) D(8)
    95 2931 3*977 a.1 (000) (2,2)
    96 3031 7*433 a.1 (000) (2,2)
    97 3048 2*3*127 #Q.6↑2 (132) Q(64)
    98 3091 11*281 a.1 (000) (2,2)
    99 3139 43*73 a.1 (000) (2,2)
    100 3147 3*1049 a.1 (000) (2,2)




Karl-Franzens University Graz, right side
Daniel C. Mayer
Explanation:

*
Web master's e-mail address:
contact@algebra.at
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