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... In general, r K,S is non-obvious and varies from 0 to r 2 + 1 (see Wingberg [6,7], Yamagishi [8], Maire [9,10,11], Labute [12], [13], Vogel [14] for some results and cases where G K,S may be free with less than r 2 + 1 generators and our forthcoming numerical results showing that many Z p -ranks can occur). ...

... Of course, dim F p (H 2 (G K,S , Z/pZ)), giving the minimal number of relations, is easily obtained only when P ⊆ S (equal to rk p (T K,S ) under Leopoldt's conjecture), which shall explain the forthcoming studies about this: [5], [6,7], [8], [11], [12], [14], [36], [44], Haberland [45], [46], El Habibi-Ziane [47] . . .. ...

... where: [11], [14] for some applications). ...

This paper is devoted to finding new examples of mild pro-p-groups as Galois groups over global fields, following the work of Labute ([6]). We focus on the Galois group GmST of the maximal T-split S-ramified pro-p-extension of a global field k. We first retrieve some facts on presentations of such a group, including a study of the local-global principle for the cohomology group H2 (GST, Fp), then we show separately in the case of function fields and in the case of number fields how it can be used to find some mild pro-p-groups.

In this paper, we study the cohomological dimen- sion of groups GS := Gal(KS/K), where KS is the maximal pro-p-extension of a number field K, unramified outside a finite set S of places of K. This dimension is well-understood only when S contains all places above p; in the case where only some of the places above p are contained in S, one can still obtain some results if KS/K contains at least one Zp-extension K1/K. Indeed, in that case, the study of the Zp((Gal(K1/K)))-module Gal(KS/K1)ab allows one to give sucient

Part I Algebraic Theory: Cohomology of Profinite Groups.- Some Homological Algebra.- Duality Properties of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa Modules.- Part II Arithmetic Theory: Galois Cohomology.- Cohomology of Local Fields.- Cohomology of Global Fields.- The Absolute Galois Group of a Global Field.- Restricted Ramification.- Iwasawa Theory of Number Fields.- Anabelian Geometry.- Literature.- Index.

In this paper we introduce a new class of Þnitely presented pro-p- groups G of cohomological dimension 2 called mild groups. If d(G);r(G) are respectively the minimal number of generators and relations of G, we give an inÞnite family of mild groups G with r(G) ï d(G) and d(G) ï 2 arbitrary. These groups can be constructed with G=(G;G) Þnite, answering a question of Kuzmin. If G = GS(p) is the Galois group of the maximal p-extension of unramiÞed outside a Þnite set of primes S and p 6= 2, we show that G is mild for a co-Þnal class of sets S, even in the case p = 2 S.

In the rst of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets. c 1997 Academic Press Limited Magma is a new software system for computational algebra, the design of which is based on the twin concepts of algebraic structure and morphism. The design is intended to provide a mathematically rigorous environment for computing with algebraic struc- tures (groups, rings, elds, modules and algebras), geometric structures (varieties, special curves) and combinatorial structures (graphs, designs and codes). The philosophy underlying the design of Magma is based on concepts from Universal Algebra and Category Theory. Key ideas from these two areas provide the basis for a gen- eral scheme for the specication and representation of mathematical structures. The user language includes three important groups of constructors that realize the philosophy in syntactic terms: structure constructors, map constructors and set constructors. The util- ity of Magma as a mathematical tool derives from the combination of its language with an extensive kernel of highly ecient C implementations of the fundamental algorithms for most branches of computational algebra. In this paper we outline the philosophy of the Magma design and show how it may be used to develop an algebraic programming paradigm for language design. In a second paper we will show how our design philoso- phy allows us to realize natural computational \environments" for dierent branches of algebra. An early discussion of the design of Magma may be found in Butler and Cannon (1989, 1990). A terse overview of the language together with a discussion of some of the implementation issues may be found in Bosma et al. (1994).

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group G
S(ℚ)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that G
S(ℚ)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension ℚS(p)/(ℚ) and we relate the cohomology of G
S(ℚ)(p) to the étale cohomology of the scheme Spec(ℤ) – S. Finally, we calculate the dualizing module.

Let p be an odd prime number and let K be an imaginary
quadratic number field whose class number is not divisible by p. For a set S of primes of K whose norm is congruent to 1 modulo p, we introduce the notion of strict circularity. We show that if S is strictly circular, then the group G(KS(p)=K) is of cohomological dimension 2 and give some explicit examples.

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group G_S(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that G_S(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension Q_S(p)/Q and we relate the cohomology of G_S(Q)(p) to the etale cohomology of the scheme Spec(Z)-S. Finally, we calculate the dualizing module.

Rings of integers of type K(π, 1) Preprints der Forschergruppe Algebraische Zykel und L-Funktionen Regensburg

- A Schmidt

Schmidt, A.: Rings of integers of type K(π, 1). Preprints der Forschergruppe Algebraische Zykel und L-Funktionen Regensburg/Leipzig Nr. 7, 2007
[V]

Circular sets of primes of imaginary quadratic number fields Preprints der Forschergruppe Algebraische Zykel und L-Funktionen Regensburg/Leipzig Nr Galois groups of number fields generated by torsion points of elliptic curves

- D Vogel
- K Wingberg

Vogel, D.: Circular sets of primes of imaginary quadratic number fields. Preprints
der Forschergruppe Algebraische Zykel und L-Funktionen Regensburg/Leipzig Nr.
5, 2006
[W]
Wingberg, K.: Galois groups of number fields generated by torsion points of elliptic
curves. Nagoya Math. J. 104 (1986), 43-53