The Oujda Research Project 2011

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

Presentations and Lectures: 27th Journées Arithmétiques Univ. Vilnius, Lithuania June 27th, 2011 The Distribution of 2nd p-Class Groups on Coclass Graphs Joint CSASC Conference Univ. Krems, Austria September 25th, 2011 Principalization Algorithm via Class Group Structure

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 coclass graph G(3,1) The interface between the coclass graphs G(3,1) and G(3,2) 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: Abelianization of type (3,3): Coclass tree T(U0) on coclass graph G(3,2) Coclass tree T(Q0) on coclass graph G(3,2) Coclass tree T(B0) on coclass graph G(3,2) with pathological branch zero Abelianization of type (9,3): Coclass tree T(A) on coclass graph G(3,2) Coclass tree T(G) on coclass graph G(3,2) 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). A summary of central ideas for the Oujda Research Project 2011 Transfers of metabelian p-groups Quadratic fields with 3-class groups of type (9,3)

Intersection of Coclass Graphs with Isoclinism Families: Stem Groups of Isoclinism Family Φ6 Metabelian 3-Groups with Abelianisation of Type (9,3) G(3,1)∩Φs s = 2,3,9,10,35,36 G(3,2)∩Φs s = 6,23,40,41,42,43 G(5,1)∩Φs s = 2,3,9,10,35,36,37,38,39