P26008-N25 Towers of p-class fields over algebraic number fields
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Final Report

Final Report English

Final report, November 2016
(Summary for public relations work)
The class field tower consists of successive extensions of a base field. Every stage is associated with a group structure. The principal target of this research project was the identification of the tower group, which corresponds to the complete tower. International cooperation enabled the final clarification of the unsolved problem for a base field with class rank two. It turned out that these towers are always of finite height, whereas they reveal unbounded growth for rank three and more. A remarkable difference appeared for complex base fields with veritable skyscrapers as towers in contrast to real base fields which are contented with rather modest heights already. The project has initiated a new era of investigating three-stage towers thereby advancing Austrian science to the forefront of international research. Fruitful impact is to be expected on the neighbor area of group theory, since the determination of the tower group is based on a new kind of periodically repeating patterns in tree diagrams. A further fundamental insight is the tendency that most components of systems of type invariants become stable with fixed values, whereas some distinguished polarized constituents remain variable and encapsulate valuable key information on the tower group. The achievements of the project have been presented at various international scientific conferences in Scotland, California, Morocco, Hungary, Slovakia and Shanghai. Joint work with mathematicians in the United States, Morocco, Japan and Australia has consolidated the international cooperation with Austria.
Scientific final report November 2016

Annual report, March 2014
In the following Annual Report 2013, I refer to the working plan [1, § 1.6, pp.16-17] of my project proposal, [1] Towers of p-class fields over algebraic number fields, in the file Proposal20130213.pdf. I also refer to certain URLs in the world wide web, in particular, to my Website for Principal Investigators of the FWF, [2] http://www.algebra.at/FWFP26008N25/index.html. In the initial stage of my project, from July to December 2013, I focussed on further development and international communication of my outstanding discoveries, providing proven evidence for the first p-class field towers of length bigger than two, for an odd prime p, namely p = 3 [1, p.16, First research line, item (1.a)]. Two coauthors were involved in these developments. Michael R. Bush in the proofs for the ground state of such towers, with relative group of smallest possible order 6561, and metabelianization with transfer kernel type (TKT) E.9. Mike F. Newman in analyzing excited states, with relative groups of order at least 177147, and TKTs E.6, 8, 9, 14. 1. Since there arose the possibility to meet M. R. Bush personally, I prepared a presentation Finite 3-groups as viewed from class field theory for the conference Groups St Andrews 2013, on August 9, abstract: http://at.yorku.ca/cgi-bin/abstract/cbhp-24, slides: [2, publ., (b.2)], conference program: http://www.groupsstandrews.org/2013/programme.shtml, and an illuminating poster, giving a succinct survey of the coclass project [2, publ., (b.3)]. The meeting with Bush enabled a fine tuning of details concerning our joint paper 3-class field towers of exact length 3 [2, publ., (a.2)]. 2. Inspired by other presentations in St Andrews, I wrote two spontaneous purely group-theoretic preprints [2, publ., (b.1)], not planned in the proposal, Normal lattice of certain metabelian p-groups and Parametrized pc-presentations of periodic sequences of 3-groups, in August. The former provides foundations for investigating TKTs in sections E and F. In the latter, I discovered a very promising characterization of mainline vertices on coclass trees which seems to be of a fairly general nature and obviously has not been recognized by other investigators up to now. Mainline Principle: The relators for pth powers of generators of mainline groups are distinguished by being independent of the class, whereas a relator of any non-mainline group contains the generator of the last non-trivial term of the lower central series as a small perturbation. This principle will be of greatest importance for the further processing of my project. 3. For the ÖMG and DMV Congress 2013, Innsbruck, I prepared my presentation 3-class field towers of exact length 3 on Sept. 24, abstract: http://www.algebra.at/DCMayerOeMG2013.pdf, slides: [2, publ., (b.2)]. 4. In December, M. R. Bush and I submitted our article 3-class field towers of exact length 3 [2, publ., (a.2)] to the Journal of Number Theory (Elsevier), and to the open source repository arXiv. To begin with a brief and compact summary and to avoid the danger of loosing priority, we restricted our announcements on the one hand to the ground state (omitting higher excited states) of TKTs in section E, corresponding to Schur σ-groups of order 6561, and on the other hand to the special TKT E.9 (skipping the related types E.6, E.8, and E.14), which is completely sufficient to provide the first rigorous disproof of the claims by Scholz/Taussky and Heider/Schmithals. For the sake of brevity, we refrained from shedding light on the caveats by Brink/Gold. 5. The most recent analysis of the results by Brink/Gold, in cooperation with Mike F. Newman, from the viewpoint of descendant trees of 3-groups, underpinned by the Mainline Principle (in 2.), was presented in my lecture Class towers and capitulation over quadratic fields at the West Coast Number Theory Conference 2013 on Dec. 18, program: http://westcoastnumbertheory.org/schedule/2013-schedule-of-talks/, slides: [2, publ., (b.2)]. As an ultimate summary of all my international presentations in 2013 [2, publ., (b.2)] and of my tetralogy [2, publ., (a.2)], I gained insight into a new strategy of pattern recognition [2, publ., (b.1)] which makes use of a systematic partial order on TKTs and TTTs of vertices on descendant trees of p-groups and will be fundamental for all further developments in my project. 6. In January 2014 I started a web presentation concerning pure quintic fields [1, p. 17, Second research line, item (3)], featuring my investigations in November and December 2013 [2, further, (1.1)]. My hypothetical classification of pure quintic fields, according to three types of principal factorizations (absolute, intermediate, and relative) and to several possibilities for the group of relative norms of units, turned out to be in perfect accordance with extensive numerical computations.

Annual report, March 2015
In the following Annual Report 2014, I refer to § 1, Scientific Aspects [1, pp.2-18], of my project proposal, Towers of p-class fields over algebraic number fields [1], in the file Proposal20130213.pdf. I also refer to the URL http://www.algebra.at/FWFP26008N25/index.html of my Website for Principal Investigators of the FWF [2]. According to [1, pp.5-6, §§ 1.3.1-1.3.2], the project consists of two research lines (RL), Two- and Three-stage towers of unramified Hilbert p-class fields, briefly (RL1), and Single-stage towers of ramified p-ring class fields, briefly (RL2). Let me begin with a target-performance comparison, based on the Working Plan and Time Schedule [1, pp.16-17, § 1.6]. Target [1, p.16, (RL1), item (1.a)] was completed in the proposed 2 months by publishing [3] Bush, Mayer, 3-class field towers of exact length 3, J. Number Theory 147 (2015), 766-777, DOI 10.1016/j.jnt.2014.08.010. (arXiv: 1312.0251v1 [math.NT] 1 Dec 2013.) An essential supplement was added by myself alone in [9, § 21.2] (see below). In further 3 months, due to several fortunate circumstances, I unexpectedly completed alone what can be done currently for the TKT in section H [10, §§ 6.2.2-6.2.3] (see below), instead of the TKTs in section F, whose investigation has been initiated in cooperation with Newman but still needs to be completed. Target [1, p.16, (RL1), item (1.b)] was completed in 3 months with results published in [10, § 6.2.1] (see below). For the target [1, p.17, (RL1), item (2.a)], the theory is now, after 2 months, completely developed with Azizi, Talbi, Derhem. Documentation is still under construction. I have communicated the numerical results with illuminating comments as a series of sequences A250236-A250242 in [4] Sloane, The On-Line Encyclopedia of Integer Sequences (OEIS), The OEIS Foundation Inc., 2014, (http://oeis.org/). See [2, Project Publications, (d.2.2)]. Target [1, p.17, (RL1), item (2.b)] was completed in 4 months and published in [5] Azizi, Zekhnini, Taous, Mayer, Principalization of 2-class groups of type (2,2,2) of biquadratic fields k = Q((p1p2q)1/2,(-1)1/2), Int. J. Number Theory (2015), DOI 10.1142/S1793042115500645. For target [1, p.17, (RL2), item (1)], though the documentation is still in progress, the theory was completely developed in 2 months, based on [6] Mayer, Quadratic p-ring spaces for counting dihedral fields, Int. J. Number Theory 10 (2014), no. 8, 2205-2242, DOI 10.1142/S1793042114500754. For target [1, p.17, (RL2), item (3)], theory, experiments, data collection and evaluation were done in 3 months, documentation being under construction. The sum of 18 months exactly corresponds to the current project state. A few words are due to my new Strategies of Dissemination [1, p.17, § 1.7]. The last two parts of my mentioned Tetralogy appeared in reverse order, due to considerable delays in the refereeing process. The fourth part, submitted Dec. 2011, [7] Mayer, The distribution of second p-class groups on coclass graphs, J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456, DOI 10.5802/jtnb842. (27th Journées Arithmétiques 2011, Vilnius University, Lithuania.) was published more than a year before the third part, submitted Aug. 2011, [8] Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415-464. Therefore, I decided not to use the Journal de Théorie des Nombres de Bordeaux (JTNB) within the frame of the Centre de Diffusion des Revues Académiques de Mathématiques (CEDRAM) any longer. In 2014, I conquered the English Wikipedia, which requires its own Wiki Markup Language, by instantly publishing four fundamental articles, (1) Descendant tree (group theory), (2) Artin transfer (group theory), (3) Principalization (algebra) and (4) p-Group generation algorithm [2, Project Publications, (d.1)], whose page view statistics prove that they have gained incredible popularity with an average of more than ten views per day and article. To transform the Wikipedia articles into three printed publications, I have selected the journal Advances in Pure Mathematics (APM) issued by the vanity press company Scientific Research Publishing (SCIRP). The first part of my New Trilogy [2, Project Publications, (a.2)] is [9] Mayer, Periodic bifurcations in descendant trees of finite p-groups, Adv. Pure Math., vol. 5, no. 4, Special Issue on Group Theory, March 2015. (arXiv: 1502.03390v1 [math.GT] 11 Feb 2015.) It contains the Wikipedia articles (1) and (4) as its foundation and striking news on periodic bifurcations [9, § 21.1], inspired by refereeing a paper by Azizi, Zekhnini, Taous, Coclass of Gal(k2(2) | k) for some fields k = Q((p1p2q)1/2,(-1)1/2) with 2-class groups of type (2,2,2), J. Algebra Appl., 2015. The second part is [10] Mayer, Index-p abelianization data of p-class tower groups, Adv. Pure Math., Special Issue on Number Theory and Cryptography, April 2015. (arXiv: 1502.03388v1 [math.NT] 11 Feb 2015.) (29th Journées Arithmétiques 2015, University of Debrecen, Hungary.) It was inspired by the 2nd version of Boston, Bush, Hajir, Heuristics for p-class towers of imaginary quadratic fields, Math. Annalen, 2015. (arXiv: 1111.4679v2 [math.NT] 10 Dec 2014.) It solves problems posed in Bartholdi, Bush, Maximal unramified 3-extensions of imaginary quadratic fields and SL2Z3, J. Number Theory 124 (2007), 159-166. The third part is under construction and will probably be called Strategy of pattern recognition via Artin transfers. It will contain the Wikipedia article (2), arithmetically structured descendant trees, inheritance, polarization and stabilization phenomena.

Annual report, March 2016
The following Annual Report 2015 concerns my stand-alone project Towers of p-class fields over algebraic number fields as described in the file Proposal20130213.pdf. I refer to various URLs in the world wide web, in particular, http://www.algebra.at/FWFP26008N25/index.html, my Website for Principal Investigators of the FWF.
In contrast to 2014, which was free of any presentations and permitted complete concentration on preparatory work for the project objectives, 2015 was full of international events where I presented the project results. 2015 yielded the harvest of 2014 in form of publications.
The first paper [1] contains foundations on descendant trees and the p-group generation algorithm, required for understanding new discoveries of periodic bifurcations with applications to covers of metabelian p-groups and identifying p-class towers.
My two weeks at the CIMPA Research School in Morocco, http://www.cimpa-icpam.org/ecoles-de-recherche/ anciens-programmes/ecoles-de-recherche-2015/liste-chronologique-des-ecoles-de/article/ theorie-des-nombres-et?lang=fr, enabled fine tuning and finishing our cooperation on [9], which will appear in IJNT.
The article [2] was inspired by marvellous computational results of Bush in 2014, lays the precise basis of the concept of IPADs of first and second order, and was presented at the 29th JA, Debrecen, http://ja2015.math.unideb.hu.
During 2015, the joint papers [7] in JNT and [8] in IJNT eventually appeared in printed form. The article [3] applies group theoretic results in [1] to number theoretic problems, sheds more light on [7] and [8], and was presented in my invited talk at the 1st ICGA, Shanghai, http://www.engii.org/ws/Home.aspx?ID=624.
Preparation and evaluation of my talk at the 22nd CSICNT, Liptovsky Jan, http://ntc.osu.cz/2015, required 2 months. Unexpectedly, I proved that real quadratic fields with capitulation types in section c have 3-stage towers. All results appeared in the conference proceedings [5] in TMMP, which deserves particular attention for the following reason. The main theorem in the paper by Shafarevich, which is most important for my project, contradicted our joint results in [8], resp. [9], on biquadratic fields containing fourth, resp. third, roots of unity and caused a lot of confusion until I discovered a fatal misprint in both, the russian original and the english translation, and published corrections in [5] and [9].
As announced, my New Trilogy, which started with [1] and [2], was completed with [4] in 2 months. The article contains a detailed theory of Artin transfers and the long desired proof of the compatibility of Artin patterns with edges of descendant trees, which thereby are endowed with additional structure. The papers [1] and [4] are devoted to pure group theory, whereas all other papers contain applications to class field theory.
The last paper [6] of 2015 is a supplement of [2], containing deeper details of computational results in a more systematic way, which were given in [2] as lurid headlines only. Although they are not joint work, the papers [2], [3], [5] and [6] would have been impossible, firstly, without Bush's permission to use his extensive numerical results of 2014 and 2015, and secondly, without my new Linux workstation with Xeon processors, supported by the Austrian Science Fund.
Further 2 months were required for computing the capitulation of all real quadratic fields with discriminants 0 < d < 5\cdot 10^7, which became possible by MAGMA version V2.21-8 (Nov. 2015), without a permanent bug up to V2.21-7.
[1] D.C. Mayer, Periodic bifurcations in descendant trees of finite p-groups, Advances in Pure Mathematics 5 (2015), no.4, 162 - 195, Special Issue on Group Theory, DOI 10.4236/apm.2015.54020.
[2] D.C. Mayer, Index-p abelianization data of p-class tower groups, Advances in Pure Mathematics 5 (2015), no.5, 286 - 313, Special Issue on Number Theory and Cryptography, DOI 10.4236/apm.2015.55029.
[3] D.C. Mayer, Periodic sequences of p-class tower groups, Journal of Applied Mathematics and Physics 3 (2015), no.7, 746 - 756, DOI 10.4236/jamp.2015.37090.
[4] D.C. Mayer, Artin transfer patterns on descendant trees of finite p-groups, Advances in Pure Mathematics 6 (2016), no.2, 66 - 104, Special Issue on Group Theory Research, DOI 10.4236/apm.2016.62008.
[5] D.C. Mayer, New number fields with known p-class tower, Tatra Mountains Mathematical Publications 64 (2015), 21 - 57, Number Theory and Cryptology '15, DOI 10.1515/tmmp-2015-0040.
[6] D.C. Mayer, Index-p abelianization data of p-class tower groups, II, J. Th\'eor. Nombres Bordeaux (2016).
[7] M.R. Bush, D.C. Mayer, 3-class field towers of exact length 3, Journal of Number Theory 147 (2015), 766 - 777, DOI 10.1016/j.jnt.2014.08.010.
[8] A. Azizi, A. Zekhnini, M. Taous, D.C. Mayer, Principalization of 2-class groups of type (2,2,2) of biquadratic fields Q( sqrt{p1p2q}, sqrt{-1} ), International Journal of Number Theory 11 (2015), no.4, 1177 - 1216, DOI 10.1142/S1793042115500645.
[9] A. Azizi, M. Talbi, M. Talbi, A. Derhem, D.C. Mayer, The group Gal( k3(2) | k ) for k=Q( sqrt{-3}, sqrt{d} ) of type (3,3), International Journal of Number Theory (2016), DOI 10.1142/S1793042116501207.


Final Report German

Final report, November 2016
(Zusammenfassung für die Öffentlichkeitsarbeit)
Der Klassenkörperturm besteht aus stufenweisen Erweiterungen eines Grundkörpers, wobei jeder Stufe eine Gruppenstruktur zugeordnet ist. Das Hauptziel dieses Forschungsprojekts war die Identifizierung der Turmgruppe, die dem gesamten Turm entspricht. Durch internationale Zusammenarbeit gelang die vollständige Aufklärung des bisher ungelösten Problems für einen Grundkörper mit Klassenrang zwei. Es stellte sich heraus, dass diese Türme stets endliche Höhe besitzen, während sie ab Rang drei ein unbeschränktes Wachstum aufweisen. Ein markanter Unterschied zeigte sich für komplexe Grundkörper mit wahrhaften Wolkenkratzern als Türmen im Gegensatz zu reellen Grundkörpern, die sich schon mit recht bescheidenen Höhen zufrieden geben. Das Projekt hat ein neues Zeitalter der Untersuchung dreistufiger Türme eingeleitet und damit die österreichische Wissenschaft an die internationale Forschungsfront gestellt. Fruchtbare Auswirkungen sind für das Nachbargebiet der Gruppentheorie zu erwarten, weil die Bestimmung der Turmgruppe auf neu entdeckten sich periodisch wiederholenden Mustern in Baumdiagrammen beruht. Eine weitere grundlegende Erkenntnis ist die Tendenz, dass sich in Systemen von Typinvarianten die meisten Bestandteile mit festen Werten stabilisieren, während einige ausgezeichnete polarisierte Komponenten veränderlich bleiben und wertvolle Schlüsselinformation über die Turmgruppe einkapseln. Die Errungenschaften des Projekts wurden an internationalen wissenschaftlichen Kongressen in Schottland, Kalifornien, Marokko, Ungarn, Slowakei und Shanghai vorgestellt. Gemeinsame Arbeit mit Mathematikern in den Vereinigten Staaten, Marokko, Japan und Australien hat die internationale Zusammenarbeit mit Österreich gefestigt.

Annual report, March 2014
Hiermit möchte ich Ihnen den Jahresbericht 2013 zu meinem dreijährigen Einzelprojekt mit dem Titel Türme von p-Klassenkörpern über algebraischen Zahlkörpern vorlegen. Das Projekt wurde am 24.06.2013 genehmigt. Die Laufzeit wurde von 01.09.2013 bis 31.08.2016 festgesetzt. Aufgrund des durch das vorgegebene Formular begrenzten Umfangs konnte ich nicht auf alle Aktivitäten eingehen, weil meine Kommentare und Erläuterungen zu den dargelegten Publikationen und Präsentationen, vor allem zu jenen mit meinen Koautoren Bush und Newman, wesentlich zum Verständnis des Projektfortschritts beitragen. Insbesondere musste ich auf die Besprechung von zwei Forschungslinien mit meinen Kooperationspartnern in Marokko verzichten. Die eine, mit Azizi, Zekhnini und Taous, betrifft neueste Erkenntnisse über bizyklische biquadratische Zahlkörper mit 2-Klassengruppen vom Typ (2,2,2) und (4,2). Die andere, mit Ayadi und Derhem, ergab interessante Neuigkeiten über zyklische kubische Zahlkörper mit 3-Klassengruppen vom Typ (3,3), (3,3,3) und (9,3). Ich werde im nächsten Projektbericht darauf eingehen.


With support from
FWFDer Wissenschaftsfonds