MAGMA 2012



Chronicle of

Research Highlights

2006 - 2016

2016
Nov 11: International Colloquium of Algebra, Number Theory, Cryptography and Information Security ANCI, Taza Invited Keynote on Recent Progress in Determining p-Class Field Towers, Taza, Morocco
Sep 08: Sensational Discovery of the First Three-Stage Tower F3K/K of 3-Class Fields over a Cyclic Cubic Field K with Conductor 28791=9*7*457
Sep 07: 3-Principalization Type and Second 3-Class Group Gal(F32K/K) of all Cyclic Cubic Fields K with 3-Class Group (3,3) and Conductor c < 100000
Aug 20: Long desired discovery of a Second 5-Class Group of Elevated Coclass cc(G52M) ≥ 3 for the Cyclic Quartic 5-Mirror Field M=Q((ζ-ζ-1)d1/2) with d = 8689
Aug 09: 5-Principalization Type and Second 5-Class Group Gal(F52M/M) of all Cyclic Quartic 5-Mirror Fields M=Q((ζ-ζ-1)d1/2), d < 10000, with 5-Class Group (5,5)
Jul 25: 2nd International Conference on Groups and Algebras ICGA, Suzhou Invited Lecture on p-Capitulation over Number Fields with p-Class Rank Two, Suzhou, China
Jun 28: 18th Postgraduate Group Theory Conference Imperial College, London Presentation Three-Stage Towers of 5-Class Fields, London, UK
Jun 25: Fork Topology, Contestants, and Shafarevich Covers cov(G,K) of metabelian 3-groups G32K with Kernel Type F and Coclass 6 over quadratic fields K
Jun 15: Annihilator Ideals of Two-Generated Metabelian p-Groups Article establishing the general templates S,W,Y and verifying the specializations L,X,Z,R,T,V
Jun 07: Precise Artin Patterns (kernels and targets of Artin transfers) AP=(TKT,TTT) of all 24476 complex quadratic fields with discriminant -107 < d, by far most extensive computation up to now
May 30: Recent Progress in Determining p-Class Field Towers Invited Lecture for the International Colloquium on Algebra, Number Theory, Cryptography and Information Security in Taza, Morocco, 2016
Apr 29: Sensational first realization of the 5-Sylow subgroup of the symmetric group of degree 25 as 2nd 5-class group Occurrence for the real quadratic field with discriminant d = 27186289
Apr 07: Sensational discovery of the first 5-CLASS TOWER OF EXACT LENGTH 3 Priority Claim Occurrence for the real quadratic fields with discriminants d = 3812377 and 19621905
Mar 16: Precise Artin Patterns (kernels and targets of Artin transfers) AP=(TKT,TTT) of all 34631 real quadratic fields with discriminant d < 108, by far most extensive computation up to now
Mar 13: Recent Progress in Determining p-Class Field Towers Abstract for the International Colloquium on Algebra, Number Theory, Cryptography and Information Security in Taza, Morocco, 2016
Feb 28: On class groups of imaginary quadratic fields Math. Rev. 3404031 of a paper by A. Wiles, in J. London Math. Soc.
Feb 18: Annual Report 2015 for the Austrian Science Fund (FWF) Report Stand-alone research project P 26008-N25
2015
Nov 24: Artin transfer patterns on descendant trees of finite p-groups Adv. Pure Math. 6 (2016), no. 1, Special Issue on Group Theory Research
Nov 12: Grunwald-Wang theorem, an effective version Math. Rev. 3368167 on a paper by S. Wang, in Sci. China Math.
Oct 18: Induced homomorphism (quotient group) Article induced homomorphism, characteristic subgroups, GI-automorphisms, functorial properties
Sep 30: New number fields with known p-class tower Tatra Mountains Math. Publ. 64 (2015), 21-57
Aug 31: 22nd Czech and Slovak International Conference on Number Theory (CSICNT) 2015, Liptovsky Jan, Slovakia Presentation New number fields with known p-class tower
Jul 21: 1st International Conference on Groups and Algebras (ICGA) 2015, Shanghai, China Presentation Invited Lecture on Periodic sequences of p-class tower groups
Jul 09: 29th Journées Arithmétiques (JA) 2015, Debrecen, Hungary Presentation Index-p abelianization data of p-class tower groups
May 18: École de Recherche CIMPA UNESCO 2015 Oujda, Morocco La théorie algorithmique des nombres (lectures and exercises)
Mar 25: Higher Newton polygons and integral bases Math. Rev. 3276340 of a paper by J. Guàrdia, J. Montes and E. Nart, in J. Number Theory
Mar 08: Descendants of <64,75> as second 2-class groups of biquadratic Dirichlet fields Q((-1)1/2,d1/2) Special instances of the cases A and D by A. Azizi and M. Taous
Feb 25: Annual Report 2014 for the Austrian Science Fund (FWF) Report Stand-alone research project P 26008-N25
Feb 18: Index-p abelianization data of p-class tower groups Adv. Pure Math. Special Issue on Number Theory and Cryptography
Jan 30: Coclass of Gal(k2(2)|k) for some fields k = Q((p1p2q)1/2,(-1)1/2) with 2-class group of type (2,2,2) Report on a paper by A. Azizi, A. Zekhnini, and M. Taous, in Journal of Algebra and its Applications
Jan 27: Periodic bifurcations in descendant trees of finite p-groups Adv. Pure Math. Special Issue on Group Theory
2014
Dec 10: The group Gal(k3(2)|k) for k = Q((-3)1/2,d1/2) of type (3,3) IJNT in cooperation with M. Talbi, A. Derhem, Mm. Talbi and A. Azizi
Dec 08: Principalization of 2-class groups of type (2,2,2) of biquadratic fields Q((p1p2q)1/2,(-1)1/2) World Scientific Int. J. Number Theory 11 (2015), no. 4, 1177-1215, DOI 10.1142/S1793042115500645, in cooperation with A. Azizi, A. Zekhnini and M. Taous.
Nov 13: Quadratic p-ring spaces for counting dihedral fields World Scientific Int. J. Number Theory 10 (2014), no. 8, 2205 - 2242, DOI 10.1142/S1793042114500754.
Oct 23: Principalization algorithm via class group structure CEDRAM J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415 - 464.
Oct 07: Signed fundamental domains for totally real number fields Math. Rev. 3198753 of a paper by F. Diaz y Diaz and E. Friedman, in Proc. London Math. Soc.
Sep 19: Endowing descendant trees T(R), R = <81,2> and <81,5>, with structure by implementation of the Artin transfers for p-groups G having G/G' = (p2,p2) or (p3,p)
Sep 18: Entering uncharted waters by construction of descendant trees T(R) with roots R = <81,2>, <81,5>, consisting of 3-groups G with abelianization G/G' = (9,9) or (27,3)
Sep 16: Completely new perspective of the population of coclass trees T2(R) with roots R = <243,6>, <243,8>, based on computations of IPADs by M. R. Bush and TKTs by D. C. Mayer
Aug 27: Structural interpretation of composite Artin transfers Article Wreath product of symmetric groups Sm and Sn with respect to {1,…,n}
Jul 31: p-group generation algorithm Article p-covering group, nucleus, allowable subgroups, orbits, step sizes
Jul 19: Complex quadratic fields of type (3,3,3) Preprint Infinite 3-class tower, transfer kernels and targets, and second 3-class group
Jul 12: On the Hilbert 2-class field tower of some abelian 2-extensions of Q Math. Rev. 3165518 of a paper by A. Azizi and A. Mouhib, in Czechoslovak Math. J.
Jul 05: Principalization in algebraic number theory Article Artin reciprocity law, commutative Artin diagram, and Galois cohomology
Jun 16: Artin transfers and structured descendant trees Article Inheritance from quotients, stabilization criteria, and pattern recognition
May 28: Strategy of pattern recognition via Artin transfers Preprint Arithmetical applications of: commutative Artin diagram, inheritance from quotients
May 21: Descendant trees of finite p-groups Article Nuclear rank, multifurcation, coclass graphs, parametrized pc-presentations
Apr 12: Principalization of 2-class groups of type (2,2,2) of biquadratic fields k = Q((-1)1/2,d1/2), d = p1p2q in cooperation with A. Azizi, A. Zekhnini, and M. Taous, in IJNT
Mar 27: Annual Report 2013 for the Austrian Science Fund (FWF) Report Stand-alone research project P 26008-N25
Feb 28: A new computational approach to ideal theory in number fields Math. Rev. 3105943 of a paper by J. Guàrdia, J. Montes, and E. Nart, in Found. Comput. Math.
Jan 31: On the strongly ambiguous classes of k|Q(i) where k = Q((2p1p2)1/2,i) Review of a paper by A. Azizi, A. Zekhnini, and M. Taous, in Asian-European Journal of Mathematics
2013
Dec 18: West Coast Number Theory 2013, Asilomar, Monterey, California Presentation Class towers and capitulation over quadratic fields
Nov 17: Classification into principal factorization types of pure quintic fields Q(D1/5) by means of metacyclic class number relations and relative norms of fundamental units
Nov 06: Using Pari/GP and Magma for the classification of pure cubic fields Q(D1/3) into principal factorization types, according to Barrucand / Cohn
Nov 03: Using ANUPQ and Magma for descendant generation of <729,9…12> of type (9,3) as an important supplement to the descendant trees of <729,13…21>
Oct 25: Units of pure quartic fields Q(p1/4) with p ≡ 7 (mod 16) Math. Rev. 3059109 of a paper by A. Aguilar-Zavoznik and M. Pineda-Ruelas, in Far East J. Math. Sci.
Oct 20: Determining the index of relative units of pure quartic fields Q(D1/4) by means of calculating the relative norms of a fundamental system of units
Oct 07: Using ANUPQ and Magma for descendant generation of <243,3> of type (3,3) to get the complete overview of transfer kernel types in section F
Sep 23: ÖMG and DMV Congress 2013, Innsbruck, Austria Presentation Abstract , 3-class field towers of exact length 3
Sep 01: Official Start of our International Research Project FWF: P 26008-N25 Towers of p-Class Fields over Algebraic Number Fields
Aug 30: French translation of the CIMPA - UNESCO Lectures "CIMPA 2015 Oujda" Resumé Théorie des Nombres Algorithmique et Applications en Cryptographie
Aug 25: Complete Proposal for the CIMPA - UNESCO Course "CIMPA 2015 Oujda" Summary Computational Number Theory and its Applications in Cryptography
Aug 22: Inspired by some presentations at "Groups St Andrews 2013": Preprint Complete Normal Lattice of metabelian p-groups G, G/G' ≅ (p,p), k(G) = 0
Aug 19: Generalization of our lecture at "Groups Saint Andrews 2013" Preprint PC-Presentations for infinite periodic sequences of 3-groups G, G/G' ≅ (9,3)
Aug 03: Groups St. Andrews 2013, Fife, Scotland Presentation Abstract , Finite 3-groups as viewed from class field theory
Jul 05: Celebrating the Thirtieth Anniversary of my Ph. D. Promotion July 5th, 1983 Periodicity and Structure of Non-Metabelian Relators, Zassenhaus Filtration
Jul 03: Non-metabelian finite 3-groups G with transfer kernel type κ(G) in Section E, cc(G) = 3 First Non-Metabelian Parametrized PC-Presentations
Jul 02: First step to find all finite 3-groups G with transfer kernel type κ(G) in Section E, cc(G) = 2 New Metabelian Parametrized PC-Presentations
Jun 17: The distribution of second p-class groups on coclass graphs CEDRAM J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401 - 456.
Jun 09: Tough computation of second layer pTKTs κ2 over complex quadratic fields of type (3,9), such as K = Q((-64671)1/2) of pTKT A.20, κ = (444;4), having a sensational κ2 = (333;2)
May 28: Discovery of further 3-towers having three stages over number fields of type (3,9), such as K = Q((-320968)1/2) of pTKT C.4 in its 1st excited state, by D. C. Mayer
May 05: First evidence of 3-class field towers having three stages over number fields of type (3,9), for example K = Q((-42567)1/2) of pTKT C.4, κ = (112;2), by D. C. Mayer
Apr 13: Criteria for sextic Pólya fields of type K = Q(ζ,D1/3), ζ = exp(2πi/3), based on investigations by Amandine Leriche (Amiens & Lille)
Apr 05: 2nd 3-class groups G32(K) of cyclic cubic fields K of type (3,9) and their connection with 2nd 3-Genus Groups g32(K)
Mar 15: Statistics of 2nd 3-class groups G32(K) of cyclic cubic fields K of type (3,3) Frequency of Abelian, Coclass 1, and Coclass 2 Groups
Mar 08: Statistics of 2nd 2-class groups G22(K) of complex & real quadratic fields K of type (2,2) Frequency of Abelian, Dihedral, Quaternion, and Semidihedral Groups
Feb 25: 2nd order IPADs τp2(K) for fixed prime p = 2, resp. p = 3, of complex quadratic fields K = Q((-1780)1/2), K = Q((-2067)1/2), resp. K = Q((-3896)1/2)
Feb 19: Complete normal lattice of three-stage 2-tower groups G23(K) of type (2,4) for complex quadratic fields K = Q((-1780)1/2) and K = Q((-2067)1/2)
Feb 17: The first 50 complex quadratic fields Q(D1/2), -2408 ≤ D ≤ -260, of type (2,4) Transfer kernel and target types of 3 layers, result: 31 (62%) single stage 2-towers
Feb 06: Discovery of a possible 3-class field tower group having three stages for a TKT in Section F namely of type F.11, first excited state, order 329, by M. F. Newman
Feb 03: Discovery of possible 3-class field tower groups having three stages for TKTs in Section F 27 of each type F.7,11,12,13, ground state, order 320, by M. F. Newman
Jan 03: Criteria for Principal Factorization Types of dihedral fields of degree 2p, α12312,γ,δ12,ε, based on integral representations of S-unit groups by Alex Bartel (Coventry)
2012
Nov 23: Transfer kernels and targets of metabelian p-groups of order p5, type (p,p), coclass 2, class 3 first commutator calculus for stem of isoclinism family Φ6, p+7 isomorphism classes
Oct 30: Discovery of further 3-class field towers having three stages, such as for k = Q((-262744)1/2) of type E.14 in its 1st excited state, by M. R. Bush and D. C. Mayer
Oct 15: Transfer kernels and targets of metabelian 7-groups of order 16807 (7,7), coclass 2, class 3 first commutator calculus for stem of isoclinism family Φ6, 14 isomorphism classes
Oct 11: Non-metabelian 3-groups of order 38, class 5, coclass 3, derived length 3 which are Schur σ-groups analysis of their lower and upper central series and pc-presentations
Sep 17: Jahrestagung der DMV, Saarbrücken, BRD Presentation Abstract
Aug 24: 3-tower has length 3 for complex quadratic fields having same TKT/TTT as k = Q((-9748)1/2) proved by N. Boston, M. R. Bush and D. C. Mayer
Aug 23: First faultless disproof of Scholz/Taussky's claim of 3-tower length 2 for k = Q((-9748)1/2) by M. R. Bush and D. C. Mayer
Aug 17: G22(k) of k = Q((-D)1/2) identified as descendant of <32,35> for D = 9380, 14980 in a series of type (2,2,2) by Lemmermeyer, realizing family #79 by Newman/O'Brien
Aug 09: Isomorphism of G32(B) and G32(k) for k = Q((-D)1/2) when k of type (3,3) 20% inherited by lifting the entire 3-class field tower, Strategic Operation on Nagasaki Day
Aug 06: G32(B) for B = Q(D1/2,(-1)1/2), Gauss-Dirichlet type, D < 3*104 211 bicyclic biquadratic (3,3) unique computation up to now, Escalation of Nuclear War on Hiroshima Day
Jul 26: Infinite pro-p Quantum Class Groups Gp(K) of finite abelianization Transfer Kernels & Targets Non-Abelian Cohen-Lenstra asymptotics: Arrigoni/Bembom/Bartholdy/Boston/Bush/Hajir
Jul 11: G22(K) for 52 fields K = Q(D1/2,(-1)1/2) with 0 < D < 104 of type (2,2,2) QCD (Quantum Chromo Dynamic) response to Zekhnini's thermonuclear bomb
Jun 14: G52(M) of 5-dual cyclic quartic mirror images M = Q((ζ-ζ-1)D1/2) of type (5,5) of 34 real, resp. 4 complex quadratic fields Q(D1/2), -16000 < D < 3500
Jun 07: G32(B) = <27,2> abelian for 79% of B = Q(D1/2,(-3)1/2) single stage 3-tower for (9,3) all 73 (100%), resp. 236/316 (75%), where Cl3Q(D1/2) = (9), resp. (3)
Jun 06: G32(B) for B = Q(D1/2,(-3)1/2), D < 5*104 389 bicyclic biquadratic (9,3) unique computation up to now (extension of Apr 07, 2012)
Jun 05: G32(B) of order 6561 for B = Q(D1/2,(-3)1/2), D = 20521, 40156 (3,3) TKT b.10 (0043) unique two examples where predicted twisting of bipolarization actually occurs
May 31: G32(B) for B = Q(D1/2,(-3)1/2), D < 5*104 930 bicyclic biquadratic (3,3) unique computation up to now (extension of Mar 07, 2012)
May 29: Transfers of metabelian p-groups Springer Monatsh. Math. 166 (2012), no. 3 - 4, 467 - 495, DOI 10.1007/s00605-010-0277-x.
May 28: Selection rules for triadic quantum class groups G = G32(B) of B = Q(D1/2,(-3)1/2) enforced by Kummer theory, Galois cohomology, automorphism groups Aut(G)
May 21: G32(B) = <729,34> for B = Q(25891/2,(-3)1/2) (3,3) TKT b.10 (0043) Ascione's non-CF-group H, derived subgroup (3,3,3,3) instead of (9,3,3)
Apr 26: International Workshop NTCCCS, Oujda, Morocco Presentation
Apr 08: G32(B) = <729,37> for B = Q(21771/2,(-3)1/2) (3,3) TKT b.10 (0043) Ascione's non-CF-group A, first triadic quantum class group of coclass 2
Apr 07: G32(B) for B = Q(D1/2,(-3)1/2), Eisenstein type, D < 3*104 213 bicyclic biquadratic (9,3) unique computation up to now, πoλεμos πατηρ παντων
Mar 07: G32(B) for B = Q(D1/2,(-3)1/2), Eisenstein type, D < 3*104 549 bicyclic biquadratic (3,3) unique computation up to now, Initialization of Nuclear War "Ianna Bimetal" (23592, 23994)
Feb 24: The second p-class group of a number field World Scientific Int. J. Number Theory 8 (2012), no. 2, 471 - 505, DOI 10.1142/S179304211250025X.
Jan 25: G32(K) = E↓4, E↓2 for K = Q(D1/2), D = 6540917, 8626101 (9,3) TKT b.15 (000;4) second layer type (9,9,9,3), resp. (9,3,3,3), instead of (9,9,3,3)
2011
Dec 29: G52(K) for K = Q(D1/2), D < 2*107 270 real (5,5) unique computation up to now
Dec 23: G52(K) for K = Q(D1/2), D > -2*106 813 complex (5,5) unique computation up to now
Nov 04: G52(K) = <3125,14> for K = Q((-89751)1/2) (5,5) TKT (123456) first realization of Taussky's fixed point principalization problem
Oct 31: G32(K) = E↓4 for K = Q(37635801/2) (9,3) TKT b.15 (000;4) first realization of descendant of Ascione's CF-group E = <243,13>
Oct 27: G32(K) = <2187,349> = H↓2 for K = Q(7003131/2) (9,3) TKT b.15 (000;4) first realization of descendant of Ascione's CF-group H = <243,14>
Sep 25: Joint CSASC Conference, Krems, Austria Presentation
Jul 29: G32(K) for K = Q(D1/2), D < 107 271 real (9,3) unique computation up to now
Jul 23: G32(K) for K = Q(D1/2), D > -106 875 complex (9,3) by far most extensive computation up to now
Jul 22: Transfer kernels and targets of metabelian 3-groups (9,3), coclass 2, class 3 first commutator calculus for branch 1 of isoclinism family Φ3
Jul 09: Transfer kernels and targets of metabelian 3-groups (9,3), coclass 3, class 3 first commutator calculus for branch 1 of isoclinism family Φ6
Jul 01: 27th Journées Arithmétiques, Vilnius, Lithuania Presentation Abstract
2010
Nov 08: Transfer kernels and targets of metabelian 5-groups of order 3125 (5,5), coclass 2, class 3 first commutator calculus for stem of isoclinism family Φ6, 12 isomorphism classes
Sep 27: Journées de Théorie des Nombres, Oujda, Morocco Presentation Reference
Jul 25: G32(K) for K = Q(D1/2), D > -106 2020 complex (3,3) by far most extensive computation up to now
Mar 31: G32(K) for K = Q(D1/2), D < 107 2576 real (3,3) by far most extensive computation up to now (extension of Dec 17, 2009)
Mar 20: Transfer kernels of type d (.043) or d* (0.43) of metabelian 3-groups (3,3), coclass ≥ 3 supplementary commutator calculus to distinguish terminal and capable vertices
Mar 18: G32(K) for K = Q(84917131/2) (3,3) TKT d*.25 (0143) first triadic quantum class group of coclass 4 over real quadratic base field
2009
Dec 17: G32(K) for K = Q(D1/2), D < 106 149 real (3,3) by far most extensive computation up to now
Dec 07: Top Down Principalization Algorithm via class group structure complete determination of 3-class groups of type (3,3,3) of unramified S3 fields
Nov 09: G32(K) for K = Q(7106521/2) (3,3) TKT b.10 (0043) first triadic quantum class group of coclass 3 over real quadratic base field
Nov 04: Transfer kernels of metabelian p-groups (p,p), coclass 1 first commutator calculus for primes p greater than or equal to 5
Nov 02: Transfer kernels of metabelian 2-groups (2,2), coclass 1 first explicit commutator calculus for exceptional prime p = 2
Sep 25: Jahrestagung der ÖMG und DMV, Graz, Austria Presentation Abstract
2008
Jan 01: G32(K) = <729,54> for K = Q(5403651/2) (3,3) TKT c.21 (0231) first realization of capable vertex by triadic quantum class group over real quadratic field
2006
Jan 30: G32(K) = <729,57> for K = Q(2147121/2) (3,3) TKT G.19 (2143) first triadic quantum class group of coclass 2 over real quadratic base field

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