The distribution of second p-class groups on coclass graphs
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 401-456.

General concepts and strategies are developed for identifying the isomorphism type of the second p-class group G=Gal(F p 2 (K)|K), that is the Galois group of the second Hilbert p-class field F p 2 (K), of a number field K, for a prime p. The isomorphism type determines the position of G on one of the coclass graphs 𝒢(p,r), r0, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p-class group Cl p (K), the position of G is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second p-class groups G for various series of number fields K having p-class groups Cl p (K) of fixed type and p{2,3,5,7}.

Des concepts et des stratégies généraux sont développés pour identifier le type d’isomorphisme du deuxième p-groupe de classes G=Gal(F p 2 (K)|K), c’est donc le groupe de Galois du deuxième p-corps de classes de Hilbert F p 2 (K), d’un corps de nombres K, pour un nombre premier p. Le type d’isomorphisme détermine la position de G sur un des graphes de coclasses 𝒢(p,r), r0, dans le sens de Eick, Leedham-Green et Newman. Il est démontré que, pour des types spéciaux du corps de base K et de son p-groupe de classes Cl p (K), la position de G est limitée à certaines branches d’arbres de coclasses admissibles par des règles de sélection. Une compréhension plus profonde, en particulier la densité de population de sommets particuliers au niveau des graphes de coclasses, est acquise en calculant la distribution effective des deuxièmes p-groupes de classes G pour diverses séries de corps de nombres K ayant des p-groupes de classes Cl p (K) de type fixe et p{2,3,5,7}.

DOI: 10.5802/jtnb.842
Classification: 11R29,  11R37,  11R11,  11R16,  11R20,  20D15
Keywords: p-class groups, p-class field tower, principalization of p-classes, quadratic fields, cubic fields, dihedral fields, metabelian p-groups, coclass graphs
Daniel C. Mayer 1

1 Naglergasse 53 8010 Graz Austria
@article{JTNB_2013__25_2_401_0,
     author = {Daniel C. Mayer},
     title = {The distribution of second $p$-class groups  on coclass graphs},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {401--456},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     doi = {10.5802/jtnb.842},
     mrnumber = {3228314},
     zbl = {1292.11126},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.842/}
}
TY  - JOUR
AU  - Daniel C. Mayer
TI  - The distribution of second $p$-class groups  on coclass graphs
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2013
DA  - 2013///
SP  - 401
EP  - 456
VL  - 25
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.842/
UR  - https://www.ams.org/mathscinet-getitem?mr=3228314
UR  - https://zbmath.org/?q=an%3A1292.11126
UR  - https://doi.org/10.5802/jtnb.842
DO  - 10.5802/jtnb.842
LA  - en
ID  - JTNB_2013__25_2_401_0
ER  - 
%0 Journal Article
%A Daniel C. Mayer
%T The distribution of second $p$-class groups  on coclass graphs
%J Journal de théorie des nombres de Bordeaux
%D 2013
%P 401-456
%V 25
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.842
%R 10.5802/jtnb.842
%G en
%F JTNB_2013__25_2_401_0
Daniel C. Mayer. The distribution of second $p$-class groups  on coclass graphs. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 401-456. doi : 10.5802/jtnb.842. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.842/

[1] M. Arrigoni, On Schur σ-groups. Math. Nachr. 192 (1998), 71–89. | MR | Zbl

[2] E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes. Abh. Math. Sem. Univ. Hamburg 5 (1927), 353–363. | MR

[3] E. Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abh. Math. Sem. Univ. Hamburg 7 (1929), 46–51. | MR

[4] 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. | Zbl

[5] J. A. Ascione, On 3-groups of second maximal class. Ph. D. Thesis, Australian National University, Canberra, 1979.

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

[7] A. Azizi et M. Taous, Determination des corps k=d,i dont le 2-groupe de classes est de type (2,4) ou (2,2,2). Rend. Istit. Mat. Univ. Trieste 40 (2008), 93–116. | MR | Zbl

[8] A. Azizi, A. 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). Prépublication, Journées de théorie des nombres, FSO, Oujda, Maroc, Septembre 2010.

[9] A. Azizi, A. Zekhnini et M. Taous, Sur la capitulation des 2-classes d’idéaux du corps 2p 1 p 2 ,i. Prépublication, Workshop International NTCCCS, FSO, Oujda, Maroc, Avril 2012.

[10] 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.

[11] L. Bartholdi and M. R. Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL 2 3 . J. Number Theory 124 (2007), 159–166. | MR | Zbl

[12] T. Bembom, The capitulation problem in class field theory. Dissertation, Georg-August-Universität Göttingen, 2012.

[13] E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary quadratic fields with Cl 2 (k)(2,2,2). J. Number Theory 103 (2003), 38–70. | MR | Zbl

[14] H. U. Besche, B. Eick, and E. A. O’Brien, The SmallGroups Library — a Library of Groups of Small Order. 2005, an accepted and refereed GAP 4 package, available also in MAGMA.

[15] N. Blackburn, On a special class of p-groups. Acta Math. 100 (1958), 45–92. | MR | Zbl

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

[17] R. Bölling, On ranks of class groups of fields in dihedral extensions over with special reference to cubic fields. Math. Nachr. 135 (1988), 275–310. | MR | Zbl

[18] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. | MR | Zbl

[19] W. Bosma, J. J. Cannon, C. Fieker, and A. Steels (eds.), Handbook of Magma functions. Edition 2.19, Sydney, 2012.

[20] N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of imaginary quadratic fields. arXiv: 1111.4679 v1 [math.NT] 20 Nov 2011.

[21] N. Boston, M. R. Bush and F. Hajir, Heuristics for p-class towers of real quadratic fields. In preparation.

[22] N. Boston and J. Ellenberg, Random pro-p groups, braid groups, and random tame Galois groups. Groups Geom. Dyn. 5 (2011), 265–280. | MR | Zbl

[23] M. Boy, On the second class group of real quadratic number fields. Dissertation, Technische Universität Kaiserslautern, 2012.

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

[25] J. R. Brink and R. Gold, Class field towers of imaginary quadratic fields. manuscripta math. 57 (1987), 425–450. | MR | Zbl

[26] M. R. Bush, Schur σ-groups of small prime power order. In preparation.

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

[28] H. Dietrich, Periodic patterns in the graph of p-groups of maximal class. J. Group Theory 13 (2010) 851–871. | MR | Zbl

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

[30] M. du Sautoy, Counting p-groups and nilpotent groups. Inst. Hautes Études Sci. Publ. Math. 92 (2001) 63–112. | MR | Zbl

[31] T. E. Easterfield, A classification of groups of order p 6 . Ph. D. Thesis, Univ. of Cambridge, 1940.

[32] B. Eick and D. Feichtenschlager, Infinite sequences of p-groups with fixed coclass. arXiv: 1006.0961 v1 [math.GR], 4 Jun 2010.

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

[34] 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.

[35] D. Feichtenschlager, Symbolic computation with infinite sequences of p-groups with fixed coclass. Dissertation, TU Braunschweig, 2010.

[36] C. Fieker, Computing class fields via the Artin map. Math. Comp. 70 (2001), no. 235, 1293–1303. | MR | Zbl

[37] G. Frei, P. Roquette, and F. Lemmermeyer, Emil Artin and Helmut Hasse. Their Correspondence 1923–1934. Universitätsverlag Göttingen, 2008. | MR

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

[39] G. Gamble, W. Nickel, and E. A. O’Brien, ANU p-Quotient — p-Quotient and p-Group Generation Algorithms, 2006, an accepted GAP 4 package, available also in MAGMA.

[40] The GAP Group, GAP — Groups, Algorithms, and Programming — a System for Computational Discrete Algebra, Version 4.4.12. Aachen, Braunschweig, Fort Collins, St. Andrews, 2008, (http://www.gap-system.org).

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

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

[43] M. Hall and J. K. Senior, The groups of order 2 n (n6). Macmillan, New York, 1964. | MR | Zbl

[44] P. Hall, The classification of prime-power groups. J. Reine Angew. Math. 182 (1940), 130–141. | MR | Zbl

[45] F.-P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. J. Reine Angew. Math. 336 (1982), 1–25. | MR | Zbl

[46] D. Hilbert, Ueber den Dirichlet’schen biquadratischen Zahlkörper. Math. Annalen 45 (1894), 309–340. | MR

[47] R. James, The groups of order p 6 (p an odd prime). Math. Comp. 34 (1980), no. 150, 613–637. | MR | Zbl

[48] Y. Kishi, The Spiegelungssatz for p=5 from a constructive approach. Math. J. Okayama Univ. 47 (2005), 1–27. | MR | Zbl

[49] H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert’s theorem 94. J. Number Theory 8 (1976), 271–279. | MR | Zbl

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

[51] C. R. Leedham-Green and S. McKay, On the classification of p-groups of maximal class. Q. J. Math. Oxford 35 (1984), 293–304. | MR | Zbl

[52] 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. | MR | Zbl

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

[54] F. Lemmermeyer, On 2-class field towers of some imaginary quadratic number fields. Abh. Math. Sem. Hamburg 67 (1997), 205–214. | MR | Zbl

[55] F. Lemmermeyer, Class groups of dihedral extensions. Math. Nachr. 278 (2005), no. 6, 679–691. | MR | Zbl

[56] The MAGMA Group, MAGMA Computational Algebra System, Version 2.19-2. Sydney, 2012, (http://magma.maths.usyd.edu.au).

[57] D. C. Mayer, Principalization in complex S 3 -fields. Congressus Numerantium 80 (1991), 73–87, Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, Canada, 1990. | MR | Zbl

[58] 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. | MR | Zbl

[59] D. C. Mayer, Transfers of metabelian p-groups. Monatsh. Math. 166 (2012), no. 3–4, 467–495, DOI 10.1007/s00605-010-0277-x. | MR | Zbl

[60] D. C. Mayer, Principalisation algorithm via class group structure. Preprint, 2011.

[61] D. C. Mayer, Metabelian 3-groups with abelianisation of type (9,3). Preprint, 2011.

[62] D. C. Mayer, The distribution of second p-class groups on coclass graphs. 27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011.

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

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

[65] K. Miyake, Algebraic investigations of Hilbert’s Theorem 94, the principal ideal theorem and the capitulation problem. Expo. Math. 7 (1989), 289–346. | MR | Zbl

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

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

[68] M. F. Newman, Groups of prime-power order. Groups — Canberra 1989, Lecture Notes in Mathematics, vol. 1456, Springer, 1990, pp. 49–62. | MR | Zbl

[69] M. F. Newman and E. A. O’Brien, Classifying 2-groups by coclass. Trans. Amer. Math. Soc. 351 (1999), 131–169. | MR | Zbl

[70] H. Reichardt, Arithmetische Theorie der kubischen Zahlkörper als Radikalkörper. Monatsh. Math. Phys. 40 (1933), 323–350. | MR | Zbl

[71] A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander. J. Reine Angew. Math. 166 (1932), 201–203. | Zbl

[72] 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. | Zbl

[73] O. Schreier, Über die Erweiterung von Gruppen II. Abh. Math. Sem. Univ. Hamburg 4 (1926), 321–346. | MR

[74] I. R. Shafarevich, Extensions with prescribed ramification points, Inst. Hautes Études Sci. Publ. Math. 18 (1963), 71–95. | Numdam | MR | Zbl

[75] O. Taussky, A remark on the class field tower. J. London Math. Soc. 12 (1937), 82–85. | MR

[76] O. Taussky, A remark concerning Hilbert’s Theorem 94. J. Reine Angew. Math. 239/240 (1970), 435–438. | MR | Zbl

Cited by Sources: