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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Periodic patterns in the graph of p-groups of maximal class,
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A new pattern in the graph of p-groups of maximal class,
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