

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 threestage 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.1617]
of my project proposal,
[1]
Towers of pclass 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 pclass 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 3groups as viewed from class field theory
for the conference Groups St Andrews 2013,
on August 9,
abstract:
http://at.yorku.ca/cgibin/abstract/cbhp24,
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
3class field towers of exact length 3
[2, publ., (a.2)].
2.
Inspired by other presentations in St Andrews, I wrote
two spontaneous purely grouptheoretic preprints
[2, publ., (b.1)],
not planned in the proposal,
Normal lattice of certain metabelian pgroups
and
Parametrized pcpresentations of periodic sequences of 3groups,
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 nonmainline group
contains the generator of the last nontrivial 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
3class 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
3class 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 σgroups 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 3groups,
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/2013scheduleoftalks/,
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 pgroups
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.218],
of my project proposal,
Towers of pclass 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.56, §§ 1.3.11.3.2],
the project consists of two research lines (RL),
Two and Threestage towers of unramified Hilbert pclass fields,
briefly (RL1), and
Singlestage towers of ramified pring class fields, briefly (RL2).
Let me begin with a targetperformance comparison,
based on the Working Plan and Time Schedule
[1, pp.1617, § 1.6].
Target [1, p.16, (RL1), item (1.a)]
was completed in the proposed 2 months by publishing
[3] Bush, Mayer,
3class field towers of exact length 3,
J. Number Theory
147
(2015),
766777,
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, §§ 6.2.26.2.3] (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 A250236A250242 in
[4] Sloane,
The OnLine 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 2class groups of type (2,2,2)
of biquadratic fields
k = Q((p_{1}p_{2}q)^{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 pring spaces for counting dihedral fields,
Int. J. Number Theory
10
(2014),
no. 8,
22052242,
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 pclass groups on coclass graphs,
J. Théor. Nombres Bordeaux
25
(2013),
no. 2,
401456,
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,
415464.
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) pGroup 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 pgroups,
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(k_{2}^{(2)}  k) for some fields
k = Q((p_{1}p_{2}q)^{1/2},(1)^{1/2})
with 2class groups of type (2,2,2),
J. Algebra Appl.,
2015.
The second part is
[10] Mayer,
Indexp abelianization data of pclass 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 pclass 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 3extensions of imaginary quadratic fields and SL_{2}Z_{3},
J. Number Theory
124
(2007),
159166.
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 standalone project
Towers of pclass 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 pgroup generation algorithm,
required for understanding new discoveries of periodic bifurcations
with applications to covers of metabelian pgroups and identifying pclass towers.
My two weeks at the CIMPA Research School in Morocco,
http://www.cimpaicpam.org/ecolesderecherche/
anciensprogrammes/ecolesderecherche2015/listechronologiquedesecolesde/article/
theoriedesnombreset?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 3stage 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 V2.218 (Nov. 2015), without a permanent bug up to V2.217.
[1]
D.C. Mayer,
Periodic bifurcations in descendant trees of finite pgroups,
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,
Indexp abelianization data of pclass 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 pclass 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 pgroups,
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 pclass tower,
Tatra Mountains Mathematical Publications
64 (2015), 21  57,
Number Theory and Cryptology '15,
DOI 10.1515/tmmp20150040.
[6]
D.C. Mayer,
Indexp abelianization data of pclass tower groups, II,
J. Th\'eor. Nombres Bordeaux
(2016).
[7]
M.R. Bush, D.C. Mayer,
3class 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 2class groups of type (2,2,2)
of biquadratic fields Q( sqrt{p_{1}p_{2}q}, 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( k_{3}^{(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 pKlassenkörpern 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 2Klassengruppen vom Typ (2,2,2) und
(4,2). Die andere, mit Ayadi und Derhem, ergab interessante Neuigkeiten
über zyklische kubische Zahlkörper mit 3Klassengruppen vom Typ (3,3),
(3,3,3) und (9,3). Ich werde im nächsten Projektbericht darauf eingehen.


