MAGMA 2012

Quantum Class Fields

and their Automorphisms

Presentations and Lectures 2012:

Objects Focussed by our Investigation.

Multiplets (N1,…,Np+1), sharing a common discriminant ,
of unramified abelian septic (p = 7), quintic (p = 5), and cubic (p = 3)
relative extensions of base objects K
give rise to (non-abelian) quantum class fields Fp2(K) and associated
quantum class groups Gp2(K) = Gal(Fp2(K)|K) of automorphisms.

Central Targets of the Project.

Research Project "MAGMA 2012" is devoted to
  • extensive applications of QuantumAlgebra's licence of MAGMA V2.19-1
    (Computational Algebra Group, School of Mathematics and Statistics,
    University of Sydney, New South Wales, Australia)
    to certain types of abelianizations G/G' of
    quantum class groups G = Gp2(K) (see Project Stages),
    aiming to determine the distribution of these
    finite metabelian p-groups on coclass graphs G(p,r), r ≥ 1,
    for small prime numbers 2 ≤ p ≤ 7,

  • a break through in the theory of metabelian pro-p-groups
    S = lim inv (Mi), associated as inverse limits
    to metabelian main lines (Mi)i of coclass trees
    forming subgraphs of the coclass graphs G(p,r), r ≥ 2,

  • analyzing the common transfer kernel type (TKT)
    and the transfer target type (TTT)
    of all populated periodic coclass sequences on the coclass graphs G(p,r), r ≥ 2,
    with the aid of parametrized presentations
    derived from pro-p-presentations of metabelian pro-p-groups S,

  • determining exact borders between vertices of
    different derived length on coclass graphs G(p,r),
    and investigating the second derived quotient G/G''
    of vertices G with derived length dl(G) = 3,
    thereby shedding light on the completely unsolved
    question of 3-stage towers of p-class fields for odd primes p.

Project Stages.

  1. Abelianization of diamond type (3,3):
    Triadic quantum class groups G32(K) on the coclass graphs G(3,r), r ≥ 1
    Bicyclic Biquadratic Base Objects of Eisenstein Type
    Bicyclic Biquadratic Base Objects of Gauss Type

  2. Abelianization of diamond type (5,5):
    Pentadic quantum class groups G52(K) on the coclass graphs G(5,r), r ≥ 1
  3. Abelianization of diamond type (7,7):
    Heptadic quantum class groups G72(K) on the coclass graphs G(7,r), r ≥ 1
  4. Theoretical foundations for any abelianization of type (p,p):
    Transfer targets and kernels of a p-adic quantum class group Gp2(K)

  5. Double layered abelianization of type (9,3):
    Triadic quantum class groups G32(K) on the coclass graphs G(3,r), r ≥ 2

Presentations and Lectures 2012:

Bibliographical References:

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[Ar1] E. Artin,
Beweis des allgemeinen Reziprozitätsgesetzes,
Abh. Math. Sem. Univ. Hamburg 5 (1927), 353 - 363.

[Ar2] E. Artin,
Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz,
Abh. Math. Sem. Univ. Hamburg 7 (1929), 46 - 51.

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[As2] J. A. Ascione,
On 3-groups of second maximal class,
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Topics in computational algebraic number theory,
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[Bb] T. Bembom,
The capitulation problem in class field theory,
Dissertation, Georg-August-Universität Göttingen, 2012.

[BEO] H. U. Besche, B. Eick, and E. A. O'Brien,
SmallGroups - a library of groups of small order,
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On a special class of p-groups,
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[Bl2] N. Blackburn,
On prime-power groups in which the derived group has two generators,
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On ranks of class groups of fields in dihedral extensions over Q with special reference to cubic fields,
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The Magma algebra system. I. The user language,
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[BCFS] W. Bosma, J. J. Cannon, C. Fieker, and A. Steels (eds.),
Handbook of Magma functions,
Edition 2.18, Sydney, 2012.

[BBH] N. Boston, M. R. Bush and F. Hajir,
Heuristics for p-class towers of imaginary quadratic fields,
arXiv:1111.4679v1 [math.NT] (2011).

[BaBu] L. Bartholdi and M. R. Bush,
Maximal unramified 3-extensions of imaginary quadratic fields and SL2Z3,
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[DEF] H. Dietrich, B. Eick, and D. Feichtenschlager,
Investigating p-groups by coclass with GAP,
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[Dt1] H. Dietrich,
Periodic patterns in the graph of p-groups of maximal class,
J. Group Theory 13 (2010), 851 - 871.

[Dt2] H. Dietrich,
A new pattern in the graph of p-groups of maximal class,
Bull. London Math. Soc. 42 (2010), 1073 - 1088.

[dS] M. du Sautoy,
Counting p-groups and nilpotent groups,
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A classification of groups of order p6,
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[EkFs] B. Eick and D. Feichtenschlager,
Infinite sequences of p-groups with fixed coclass
(preprint 2010).

[ELNO] B. Eick, C. R. Leedham-Green, M. F. Newman, and E. A. O'Brien,
On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2
(preprint 2011).

[Fi] C. Fieker,
Computing class fields via the Artin map,
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[Fu] Ph. Furtwängler,
Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper,
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[GAP] The GAP Group,
GAP - Groups, Algorithms, and Programming, Version 4.4.12,
Aachen, Braunschweig, Fort Collins, St. Andrews, 2008,

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Ranks of 3-class groups of non-Galois cubic fields,
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[Lm] F. Lemmermeyer,
Class groups of dihedral extensions,
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[MAGMA] The MAGMA Group,
MAGMA Computational Algebra System, Version 2.19-1,
Sydney, 2012,

[Ma] D. C. Mayer,
Multiplicities of dihedral discriminants,
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[Ma1] D. C. Mayer,
Principalization in complex S3-fields,
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[Ma2] D. C. Mayer,
Transfers of metabelian p-groups,
Monatsh. Math. 166 (2012), no. 3 - 4, 467 - 495, DOI 10.1007/s00605-010-0277-x.

[Ma3] D. C. Mayer,
The second p-class group of a number field,
Int. J. Number Theory 8 (2012), no. 2, 471 - 505, DOI 10.1142/S179304211250025X.

[Ma4] D. C. Mayer,
Principalization algorithm via class group structure
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[Ma5] D. C. Mayer,
The distribution of second p-class groups on coclass graphs,
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27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011.

[mL] C. McLeman,
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[Mi] R. J. Miech,
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[My] K. Miyake,
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[Ne2] B. Nebelung,
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und Anwendung auf das Kapitulationsproblem
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[Nm] M. F. Newman,
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[PARI] The PARI Group,
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[Re] H. Reichardt,
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[SnKw] J.-J. Son and S.-H. Kwon,
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[Su] H. Suzuki,
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[Ta1] O. Taussky,
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[Ta2] O. Taussky,
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[Yo] E. Yoshida,
On the 3-class field tower of some biquadratic fields,
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