Oujda 2011



The Oujda Research Project 2011

Galois Groups of p-Class Field Towers as p-Groups on Coclass Graphs



Presentations and Lectures:



Presentation


Most recent partial project "Nine By Three"

Inspired by the 27th Journées Arithmétiques, University of Vilnius, Lithuania,
we succeeded in finishing the construction, analysis, and statistical evaluation of
the second 3-class groups G32(K) of all quadratic fields K = Q(D1/2)
having a 3-class group of type (9,3) and discriminant -106 < D < 107,
on July 30, 2011, by means of a veritable "tour de force" of computing:
Nine By Three (Project),
Metabelian 3-Groups with Two Generators (Theory).
This was the second successful application of our incredibly powerful
Top-Down Principalization Algorithm Via Class Group Structure,
presented in detail at the Joint CSASC Conference,
of the Chekh Republic, Slovakia, Austria, Slovenia, and Catalunya,
University of Krems, Austria, September 25, 2011.



Coclass theory of finite p-groups, infinite pro-p-groups, and uniserial p-adic space groups

The idea of using the coclass cc(G) of a finite p-group G
as the primary invariant for classification
is due to Mike F. Newman and Charles R. Leedham-Green [LgNm].

For a given prime p,
Newman and Leedham-Green have defined the structure of a directed graph G(p)
on the set of all isomorphism classes of finite p-groups.
Two vertices are connected by an edge G → H
if G is isomorphic to the last lower central quotient H/γc(H)
where c = cl(H) denotes the nilpotency class of H.

If the condition |H| = p |G| is imposed on the edges,
G(p) is partitioned into countably many disjoint subgraphs G(p,r), r ≥ 0,
called coclass graphs of p-groups G of coclass r = cc(G) = n-cl(G)
where |G| = pn.

A coclass graph G(p,r)
is a forest of finitely many coclass trees T(Gi) with roots Gi,
each with a single infinite mainline having a pro-p-group Si of coclass r as its inverse limit,
and additionally contains finitely many sporadic groups outside of coclass trees, e.g.

The origins of coclass theory go back to the first computer aided constructions of p-groups
by means of the nilpotent quotient algorithm and p-group extension or generation algorithm.
One of the first computations of this kind is due to Judith Ascione, George Havas, and Leedham-Green.
In [AHL], [As],
they determined the structure of the coclass graph G(3,2) up to order 310:

Newman, Eamonn O'Brien, Leedham-Green, and Susan McKay, [NmOb], [LgMk],
investigated the connection between finite p-groups, infinite pro-p-groups, and uniserial p-adic space groups,
and proved the coclass theorems.
Bettina Eick, Leedham-Green, Heiko Dietrich, and Dörte Feichtenschlager, [EkLg], [DEF], [EkFs],
are investigating periodicity phenomena in the coclass trees T(Gi)
of finite p-groups of coclass r,
which lead to parametrized presentations of infinite series of p-groups.
Calculating the transfers from a metabelian p-group G to its maximal subgroups
with the aid of its presentation
immediately yields the solution of the principalization problem for
the second p-class group Gp2(K) = Gal(Fp2(K)|K) [Ma], [Ma1]
of an algebraic number field K
with p-class group isomorphic to the abelianization G/G'.



Second and higher p-class groups of algebraic number fields

A central aim of the present research project
is to determine which metabelian (resp. non-metabelian) p-groups
can occur as the second p-class group Gp2(K) = Gal(Fp2(K)|K)
(resp. as the higher p-class groups Gpn(K) = Gal(Fpn(K)|K) with n ≥ 3)
of certain base fields K.
Due to the connection between the transfers
from the metabelian p-group Gp2(K) to its maximal normal subgroups
and the principalization of ideal classes of the base field K
in unramified cyclic extensions of K of relative degree p,
this problem can be solved essentially by determining the principalization type of K
and comparing with the transfer kernel types of metabelian p-groups G
having an abelianization G/G' isomorphic to the p-class group of K.



Participants of the Oujda Research Project 2011

The present research project
is a joint enterprise of professors, doctors, and dissertants
of the Faculté des Sciences d'Oujda (FSO)
at the Université Mohammed Premier in Oujda, Morocco,
and scientific collaborators in Fez, Taza, Casablanca, Melilla, Abha, and Graz, Austria.
It is devoted to the investigation of
p-class field towers over algebraic number fields,
the structure of their Galois groups,
and the principalization of ideal classes of order a power of p.
The project is based on
algebraic number theory, Kummer theory, Iwasawa theory,
class field theory, and genus field theory
and aims to close up to the research frontier of the
theory of pro-p-groups and the coclass theory of finite p-groups,
which are of central interest in current theoretical and computational research.



Research of the project participants

In their previous research,
the participants of the project have gained considerable experience
in the principalization problem of various base fields, such as
quadratic fields
(Raymond Couture and Aïssa Derhem [Ki2], [CtDh], [BjSn], [Ma] for p=2,
and Mohamed Talbi and Daniel C. Mayer [Tl], [Ma1], [Ma2] for p=3,5),
cyclic cubic fields
(Aïssa Derhem [Dh] for p=2, and Mohammed Ayadi [Ay] for p=3),
cyclic or bicyclic biquadratic fields
(Abdelmalek Azizi, Ali Mouhib, Mohammed Talbi, Mohammed Taous, and Abdelkader Zekhnini
[Az], [AzMh], [AzTb], [AzTs], [AZT] for p=2),
and non-abelian sextic fields
(Moulay Chrif Ismaïli and Rachid El Mesaoudi [Is], [IsMe] for p=3).
Other activities concern Voronoi's algorithms
for computing lattice minima and differential principal factors
in quadratic and cubic orders
(Ouafae Lahlou and Mohamed El Hassani Charkani [LhCh] and
Daniel C. Mayer [Ma4] for p=3, [Ma3] for p=2).



Summary, preprints, and computational details

The Oujda Research Project 2011 will start with
class field theoretical applications of p-groups G
on the well-known coclass graphs G(p,1) for p = 2,3,5,
having abelianisations G/G' of type (p,p).
However, it will then continue by a break through
to the coclass graphs G(p,2) for p = 2,3,5,
which need a great deal of investigation for p = 3,5,
and to G(p,r) for p = 2,3 and r ≥ 3.
The types of abelianizations addressed by the continuation
are mainly (p,p), (p2,p), and (p,p,p).



Intersection of Coclass Graphs
with Isoclinism Families:



Discussion





Bibliographical References:

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

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

[As] Judith Ann Ascione,
On 3-groups of second maximal class,
Ph.D. Thesis, Australian National University, Canberra, 1979,
and Bull. Austral. Math. Soc. 21 (1980), 473 - 474.

[AHL] J. A. Ascione, George Havas, and Charles 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.

[Ay] Mohammed Ayadi,
Sur la capitulation des 3-classes d'idéaux d'un corps cubique cyclique,
Thèse de doctorat, Université Laval, Québec, 1995.

[Az] Abdelmalek Azizi,
Sur la capitulation des 2-classes d'idéaux de Q(d1/2,i),
Thèse de doctorat, Université Laval, Québec, 1993,
et C. R. Acad. Sci. Paris, Sér. I, 325 (1997), 127 - 130.

[AzMh] A. Azizi et Ali Mouhib,
Capitulation des 2-classes d'idéaux de Q(21/2,d1/2) où d est un entier naturel sans facteurs carrés,
Acta Arith. 109 (2003), no. 1, 27 - 63.

[AzTb] A. Azizi et Mohammed Talbi,
Capitulation des 2-classes d'idéaux de certains corps biquadratiques cycliques,
Acta Arith. 127 (2007), no. 3, 231 - 248.

[AzTs] A. Azizi et Mohammed Taous,
Capitulation des 2-classes d'idéaux de k=Q((2p)1/2,i),
Acta Arith. 131 (2008), no. 2, 103 - 123.

[AZT] A. Azizi, Abdelkader Zekhnini et M. Taous,
Capitulation dans le corps des genres de certain corps de nombres biquadratique imaginaire dont le 2-groupe des classes est de type (2,2,2),
preprint, 2010.

[BaCo1] Pierre Barrucand and Harvey Cohn,
A rational genus, class number divisibility, and unit theory for pure cubic fields,
J. Number Theory 2 (1970), 7 - 21.

[BaCo2] Pierre Barrucand and Harvey Cohn,
Remarks on principal factors in a relative cubic field,
J. Number Theory 3 (1971), 226 - 239.

[BWB] Pierre Barrucand, Hugh C. Williams, and L. Baniuk,
A computational technique for determining the class number of a pure cubic field,
Math. Comp. 30 (1976), no. 134, 312 - 323.

[BjSn] Elliot Benjamin and C. Snyder,
Real quadratic number fields with 2-class group of type (2,2),
Math. Scand. 76 (1995), 161 - 178.

[Bw] W. E. H. Berwick,
On cubic fields with a given discriminant,
Proc. London Math. Soc., Ser. 2, 23 (1925), 359-378.

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

[Br] James R. Brink,
The class field tower for imaginary quadratic number fields of type (3,3),
Dissertation, Ohio State Univ., 1984.

[BrGo] James R. Brink and Robert Gold,
Class field towers of imaginary quadratic fields,
manuscripta math. 57 (1987), 425 - 450.

[CtDh] Raymond Couture et Aïssa Derhem,
Un problème de capitulation,
C. R. Acad. Sci. Paris, Série I, 314 (1992), 785 - 788.

[Dh] Aïssa Derhem,
Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques,
Thèse de doctorat, Université Laval, Québec, 1988.

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

[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.

[EnTu1] Veikko Ennola and Reino Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), 495 - 518.

[EnTu2] Veikko Ennola and Reino Turunen,
On cyclic cubic fields,
Math. Comp. 45 (1985), 585 - 589.

[Fu1] Ph. Furtwängler,
Über das Verhalten der Ideale des Grundkörpers im Klassenkörper,
Monatsh. Math. Phys. 27 (1916), 1 - 15.

[Fu2] Ph. Furtwängler,
Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper,
Abh. Math. Sem. Univ. Hamburg 7 (1929), 14 - 36.

[Fu3] Ph. Furtwängler,
Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper,
J. Reine Angew. Math. 167 (1932), 379 - 387.

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

[Gr] Marie-Nicole Gras,
Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q,
J. reine angew. Math. 277 (1975), 89 - 116.

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

[Ha3] Helmut Hasse,
Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage,
Math. Z. 31 (1930), 565 - 582.

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

[Hi1] David Hilbert,
Über den Dirichlet'schen biquadratischen Zahlkörper,
Math. Annalen 45 (1894), 309 - 340.

[Hi2] David Hilbert,
Die Theorie der algebraischen Zahlkörper,
Jber. der D. M.-V. 4 (1897), 175 - 546.

[Ho] Taira Honda,
Pure cubic fields whose class numbers are multiples of three,
J. Number Theory 3 (1971), 7 - 12.

[Is] Moulay Chrif Ismaïli,
Sur la capitulation des 3-classes d'idéaux de la clôture normale d'un corps cubique pur,
Thèse de doctorat, Université Laval, Québec, 1992.

[IsMe1] Moulay Chrif Ismaïli and Rachid El Mesaoudi,
Sur la divisibilité exacte par 3 du nombre de classes de certain corps cubiques purs,
Ann. Sci. Math. Québec 25 (2001), no. 2, 153 - 177.

[IsMe] Moulay Chrif Ismaïli and Rachid El Mesaoudi,
Sur la capitulation des 3-classes d'idéaux de la clôture normale de certaines corps cubiques purs,
Ann. Sci. Math. Québec 29 (2005), no. 1, 49 - 72.

[Ki1] Hershy Kisilevsky,
Some results related to Hilbert's theorem 94,
J. number theory 2 (1970), 199 - 206.

[Ki2] Hershy Kisilevsky,
Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94,
J. number theory 8 (1976), 271 - 279.

[Ku1] Tomio Kubota,
Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen biquadratischen Zahlkörpers,
Nagoya Math. J. 6 (1953), 119 - 127.

[Ku2] Tomio Kubota,
Über den bizyklischen biquadratischen Zahlkörper,
Nagoya Math. J. 10 (1956), 65 - 85.

[Kd] S. Kuroda,
Über den Dirichletschen Körper,
J. Fac. Sci. Imp. Univ. Tokyo, Sec. I, Vol. 4 (1943), Part 5, 383 - 406.

[LhCh] O. Lahlou et M. Charkani El Hassani,
Arithmétique d'une famille de corps cubiques,
C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 371 - 376.

[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.

[Mt] J. Martinet,
Sur l'arithmétique des extensions galoisiennes à groupe de galois diédral d'ordre 2p,
Ann. Inst Fourier, Grenoble 19 (1963), 1 - 80.

[MtPn] J. Martinet et J.-J. Payan,
Sur les extensions cubiques non-Galoisiennes de rationels et leur clôture Galoisienne,
J. reine angew. Math. 228 (1965), 15 - 37.

[Ma] Daniel C. Mayer,
Transfers of metabelian p-groups,
Monatsh. Math. (2010), DOI 10.1007/s00605-010-0277-x.

[Ma1] Daniel C. Mayer,
The second p-class group of a number field,
Int. J. Number Theory (2010), DOI 10.1142/S179304211250025X.

[Ma2] Daniel C. Mayer,
Principalization algorithm via class group structure,
preprint, 2011.

[Ma3] Daniel C. Mayer,
Lattice minima and units in real quadratic number fields,
Publicationes Mathematicae Debrecen
39 (1991), 19-86.

[Ma4] Daniel C. Mayer,
Differential principal factors and units in pure cubic number fields,
Dept. of Math., Univ. Graz, 1989.

[Ma5] Daniel C. Mayer,
Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), no. 198, 831-847, and supplements section S55-S58.

[Ma6] Daniel 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.

[Ma7] Daniel C. Mayer,
Discriminants of metacyclic fields,
Canad. Math. Bull. 36(1) (1993), 103-107.

[Ma8] Daniel C. Mayer,
Classification of dihedral fields,
Dept. of Computer Science, Univ. of Manitoba, 1991.

[Ma9] Daniel C. Mayer,
List of discriminants dL<200000 of totally real cubic fields L, arranged according to their multiplicities m and conductors f,
Dept. of Computer Science, Univ. of Manitoba, 1991.

[Ma10] Daniel C. Mayer,
Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106,
Univ. Graz, Computer Centre, 2002.

[Ma11] Daniel C. Mayer,
Class Numbers and Principal Factorizations of Families of Cyclic Cubic Fields with Discriminant d < 1010,
Univ. Graz, Computer Centre, 2002.

[Ma12] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of all Quadratic Fields with Discriminant -50000 < d < 0 and 3-Class Group of Type (3,3)
,
Univ. Graz, Computer Centre, 2003

[Ma13] Daniel C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of selected Quadratic Fields with Discriminant -200000 < d < -50000 and 3-Class Group of Type (3,3)
,
Univ. Graz, Computer Centre, 2004

[Ma14] Daniel C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
Univ. Graz, 2006.

[Ma15] Daniel C. Mayer,
3-Capitulation over Quadratic Fields with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2006.

[Ma16] Daniel C. Mayer,
Quadratic p-ring spaces for counting dihedral fields,
Dept. of Computer Science, Univ. of Manitoba, 2009.

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

[Mo] Nicole Moser,
Unités et nombre de classes d'une extension Galoisienne diédrale de Q,
Abh. Math. Sem. Univ. Hamburg 48 (1979), 54 - 75.

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

[Ne2] Brigitte 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.

[NmOb] M. F. Newman and E. A. O'Brien,
Classifying 2-groups by coclass,
Trans. Amer. Math. Soc. 351 (1999), 131 - 169.

[Pa] Charles J. Parry,
Bicyclic Bicubic Fields,
Canad. J. Math. 42 (1990), no. 3, 491 - 507.

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

[Sm] Bodo Schmithals,
Kapitulation der Idealklassen und Einheitenstruktur in Zahlkörpern,
J. Reine Angew. Math. 358 (1985), 43 - 60.

[So1] Arnold Scholz,
Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,
J. Reine Angew. Math. 166 (1932), 201-203

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

[SoTa] Arnold Scholz und Olga 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] Otto Schreier,
Über die Erweiterung von Gruppen. I,
Monatsh. Math. Phys. 34 (1926), 165 - 180.

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

[Tl] Mohamed Talbi,
Capitulation des 3-classes d'idéaux dans certains corps de nombres,
Thèse de doctorat, Université Mohammed Premier, Oujda, Morocco, 2008.

[Ta1] Olga Taussky,
Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper,
J. Reine Angew. Math. 168 (1932), 193 - 210.

[Ta2] Olga Taussky,
A remark on the class field tower,
J. London Math. Soc. 12 (1937), 82 - 85.

[Vo1] Georgij F. Voronoi,
O celykh algebraicheskikh chislakh zavisyashchikh ot kornya uravneniya tretei stepeni
(On the algebraic integers derived from a root of a third degree equation),
Master's thesis, 1894, St. Peterburg (Russian).

[Vo2] Georgij F. Voronoi,
Ob odnom obobshchenii algorifma nepreryvnykh drobei
(On a generalization of the algorithm of continued fractions),
Doctoral Dissertation, 1896, Warsaw (Russian).

[Wa1] Hideo Wada,
On cubic Galois extensions of Q( (-3)1/2 ),
Proc. Japan Acad. 46 (1970), 397 - 400.

[Wa2] Hideo Wada,
A table of ideal class groups of imaginary quadratic fields,
Proc. Japan Acad. 46 (1970), 401 - 403.

[Wa3] Hideo Wada,
A table of fundamental units of purely cubic fields,
Proc. Japan Acad. 46 (1970), 1135 - 1140.

[Wi3] Hugh C. Williams,
Determination of principal factors in Q(D1/2) and Q(D1/3),
Math. Comp. 38 (1982), no. 157, 261 - 274.

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

Back to Algebra