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:

[Ag] M. Arrigoni,
On Schur σ-groups,
Math. Nachr. 192 (1998), 71 - 89.

[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.

[AHL] J. A. Ascione, G. Havas, and C. R. Leedham-Green,
A computer aided classification of certain groups of prime power order,
Bull. Austral. Math. Soc. 17 (1977), 257 - 274, Corrigendum 317 - 319, Microfiche Supplement p.320.

[As1] J. A. Ascione,
On 3-groups of second maximal class,
Ph.D. Thesis, Austral. National Univ., Canberra, 1979.

[As2] J. A. Ascione,
On 3-groups of second maximal class,
Bull. Austral. Math. Soc. 21 (1980), 473 - 474.

[Bg] G. Bagnera,
La composizione dei gruppi finiti il cui grado è la quinta potenza di un numero primo,
Ann. di Mat. (Ser. 3) 1 (1898), 137 - 228.

[Be] K. Belabas,
Topics in computational algebraic number theory,
J. Théor. Nombres Bordeaux 16 (2004), 19 - 63.

[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,
2005, a refereed GAP 4 package.

[Bl1] N. Blackburn,
On a special class of p-groups,
Acta Math. 100 (1958), 45 - 92.

[Bl2] N. Blackburn,
On prime-power groups in which the derived group has two generators,
Proc. Camb. Phil. Soc. 53 (1957), 19 - 27.

[Boe] R. Bölling,
On ranks of class groups of fields in dihedral extensions over Q with special reference to cubic fields,
Math. Nachr. 135 (1988), 275 - 310.

[BCP] W. Bosma, J. Cannon, and C. Playoust,
The Magma algebra system. I. The user language,
J. Symbolic Comput. 24 (1997), 235 - 265.

[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,
J. Number Theory 124 (2007), 159 - 166.

[ChFt] S. M. Chang and R. Foote,
Capitulation in class field extensions of type (p,p),
Can. J. Math. 32 (1980), no. 5, 1229 - 1243.

[DEF] H. Dietrich, B. Eick, and D. Feichtenschlager,
Investigating p-groups by coclass with GAP,
Computational group theory and the theory of groups, pp. 45 - 61, Contemp. Math. 470, AMS, Providence, RI, 2008.

[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,
Inst. Hautes Études Sci. Publ. Math. 92 (2001), 63 - 112.

[Ef] T. E. Easterfield,
A classification of groups of order p6,
Ph. D. Thesis, Univ. of Cambridge, 1940.

[EkLg] B. Eick and C. Leedham-Green,
On the classification of prime-power groups by coclass,
Bull. London Math. Soc. 40 (2) (2008), 274 - 288.

[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,
Math. Comp. 70 (2001), no. 235, 1293 - 1303.

[Fu] Ph. Furtwängler,
Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper,
Abh. Math. Sem. Univ. Hamburg 7 (1929), 14 - 36.

[GAP] The GAP Group,
GAP - Groups, Algorithms, and Programming, Version 4.4.12,
Aachen, Braunschweig, Fort Collins, St. Andrews, 2008, http://www.gap-system.org

[Ge] F. Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307 - 322.

[Gr] G. Gras,
Sur les l-classes d'idéaux des extensions non galoisiennes de degré premier impair l
à la clôture galoisienne diédrale de degré 2l
,
J. Math. Soc. Japan 26 (1974), 677 - 685.

[Hl] Ph. Hall,
The classification of prime-power groups,
J. reine angew. Math. 182 (1940), 130 - 141.

[HeSm] F.-P. Heider und B. Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25.

[Jm] R. James,
The groups of order p6 (p an odd prime),
Math. Comp. 34 (1980), no. 150, 613 - 637.

[KoVe] H. Koch und B. B. Venkov,
Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers,
Astérisque, 24 - 25 (1975), 57 - 67.

[LgMk] C. R. Leedham-Green and S. McKay,
The structure of groups of prime power order,
London Math. Soc. Monographs, New Series, 27, Oxford Univ. Press, 2002.

[LgNm] C. R. Leedham-Green and M. F. Newman,
Space groups and groups of prime power order I,
Arch. Math. 35 (1980), 193 - 203.

[Lm] F. Lemmermeyer,
Class groups of dihedral extensions,
Math. Nachr. 278 (2005), no. 6, 679 - 691.

[MAGMA] The MAGMA Group,
MAGMA Computational Algebra System, Version 2.19-1,
Sydney, 2012, http://magma.maths.usyd.edu.au

[Ma] D. C. Mayer,
Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), no. 198, 831-847, supplements section S55-S58, DOI 10.2307/2153221.

[Ma1] D. C. Mayer,
Principalization in complex S3-fields,
Congressus Numerantium 80 (1991), 73 - 87,
Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, 1990.

[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
(preprint 2011).

[Ma5] D. C. Mayer,
The distribution of second p-class groups on coclass graphs,
J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401 - 456.
27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011.

[mL] C. McLeman,
p-tower groups over quadratic imaginary number fields,
Ann. Sci. Math. Québec 32 (2008), no. 2, 199 - 209.

[Mi] R. J. Miech,
Metabelian p-groups of maximal class,
Trans. Amer. Math. Soc. 152 (1970), 331 - 373.

[My] K. Miyake,
Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem,
Expo. Math. 7 (1989), 289 - 346.

[Ne1] B. Nebelung,
Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem
,
Inauguraldissertation, Band 1, Univ. zu Köln, 1989.

[Ne2] B. Nebelung,
Anhang zu Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem
,
Inauguraldissertation, Band 2, Univ. zu Köln, 1989.

[Nm] M. F. Newman,
Groups of prime-power order
Groups - Canberra 1989, Lecture Notes in Mathematics, vol. 1456, 1990, pp. 49 - 62.

[PARI] The PARI Group,
PARI/GP, Version 2.3.4
Bordeaux, 2008, http://pari.math.u-bordeaux.fr

[Re] H. Reichardt,
Arithmetische Theorie der kubischen Zahlkörper als Radikalkörper,
Monatsh. Math. Phys. 40 (1933), 323 - 350.

[So1] A. Scholz,
Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,
J. reine angew. Math. 166 (1932), 201 - 203.

[So2] A. Scholz,
Idealklassen und Einheiten in kubischen Körpern,
Monatsh. Math. Phys. 40 (1933), 211 - 222.

[SoTa] A. Scholz und O. Taussky,
Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper:
ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm,
J. reine angew. Math. 171 (1934), 19 - 41.

[Sr1] O. Schreier,
Über die Erweiterung von Gruppen. I,
Monatsh. Math. Phys. 34 (1926), 165 - 180.

[Sr2] O. Schreier,
Über die Erweiterung von Gruppen. II,
Hamburg. Sem. Abh. 4 (1926), 321 - 346.

[SnKw] J.-J. Son and S.-H. Kwon,
On the principal ideal theorem,
J. Korean Math. Soc. 44 (2007), no. 4, 747 - 756.

[Su] H. Suzuki,
A generalization of Hilbert's Theorem 94,
Nagoya Math. J. 121 (1991), 161 - 169.

[Ta1] O. Taussky,
A remark on the class field tower,
J. London Math. Soc. 12 (1937), 82 - 85.

[Ta2] O. Taussky,
A remark concerning Hilbert's Theorem 94,
J. Reine Angew. Math. 239/240 (1970), 435 - 438.

[Yo] E. Yoshida,
On the 3-class field tower of some biquadratic fields,
Acta Arith. 107 (2003), no. 4, 327 - 336.

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