Free pro-p groups as galois groups over ℚ(p)(t)

Letp be a prime and let ℚ(p) denote the maximalp-extension of ℚ. We prove that for every primep, the free pro-p group on countably many generators is realizable as a regular extension of ℚ(p)(t). As a consequence, if ℚ _nil denotes the maximal nilpotent extension of ℚ, then every finite nilpotent group is realizable as a regular extension of ℚ _nil (t).     

Représentations <Emphasis Type="Italic">p</Emphasis>-adiques de GL<Subscript>2</Subscript>(L) et Catégories Dérivées

We construct some locally ℚ _p -analytic representations of GL₂(L), L a finite extension of ℚ _p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel’d’s upper half space has a structure of rank 2 admissible filtered (φ, N)-module. Finally, we prove that this filtered module is associated, via Fontaine’s theory, to the initial Galois representation.     

Onp-class groups of cyclic extensions of prime degreep of certain cyclotomic fields

Letp be an odd prime number, and letK be a cyclic extension of ℚ(ζ) of degreep, where ζ is a primitivep-th root of unity. LetC _K be thep-class group ofK, and letr _K be the minimal number of generators ofC _K ^1−σ as a module over Gal(K/ℚ(ζ)), were σ is a generator of Gal(K/ℚ(ζ)). This paper shows how likely it is forr _K = 0, 1, 2, … whenp=3, 5, or 7, and describes the obstacle to generalizing these results to regular primesp>7.     

Q-admissibility of certain nonsolvable groups

A finite groupG is calledQ-admissible if there exists a finite dimensional central division algebra overQ, containing a maximal subfield which is a Galois extension ofQ with Galois group isomorphic toG. It is proved thatS ₅ ⁻ , one of the two nontrivial central extensions ofS ₅ byZ/2Z, isQ-admissible. As a consequence of that result and previous results of Sonn and Stern, every finite Sylow-metacyclic group, havingA ₅ as a composition factor, isQ-admissible. This paper is part of a M.Sc. thesis written at the Technion — Israel Institute of Technology, under the supervision of Professor J. Sonn, whom the author wishes to thank for his valuable guidance.     

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ^∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ^∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations. T. Dokchitser is supported by a Royal Society University Research Fellowship.     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Weyl groups as Galois groups of a regular extension of the field ℚ

For a finite group G, Gal_T(G) denotes the property that there exists a regular Galois extension of the rational function field ℚ(T) over the field of rationals ℚ, with a Galois group G. This property is established to be satisfied by all Weyl groups except the type F₄, from which it follows that it holds also for Chevalley groups C₃(2) and D₄(2). Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 311-315, May-June, 1995.     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

The group of endo-permutation modules

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of ℚ⊗D(-) viewed as a functor, which turns out to be a simple functor S _{E, ℚ}, indexed by the elementary group E of order p ² and the trivial Out(E)-module ℚ. Finally we describe a rather strange exact sequence relating ℚ⊗D(P), ℚ⊗B(P), and ℚ⊗R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of ℚP-modules. Oblatum 6-VII-1998 & 27-V-1999 / Published online: 22 September 1999     

Some Elements of Finite Order in K2Q

Let K2 be the Milnor functor and let Фn （x）∈ Q[X] be the n-th cyclotomic polynomial. Let Gn（Q） denote a subset consisting of elements of the form {a, Фn（a）}, where a ∈ Q^＊ and {, } denotes the Steinberg symbol in K2Q. J. Browkin proved that Gn（Q） is a subgroup of K2Q if n = 1,2, 3, 4 or 6 and conjectured that Gn（Q） is not a group for any other values of n. This conjecture was confirmed for n =2^T 3S or n = p^r, where p ≥ 5 is a prime number such that h（Q（ζp）） is not divisible by p. In this paper we confirm the conjecture for some n, where n is not of the above forms, more precisely, for n = 15, 21,33, 35, 60 or 105.     

Independent systems of representatives in weighted graphs

The following conjecture may have never been explicitly stated, but seems to have been floating around: if the vertex set of a graph with maximal degree Δ is partitioned into sets V _i of size 2Δ, then there exists a coloring of the graph by 2Δ colors, where each color class meets each V _i at precisely one vertex. We shall name it the strong 2Δ-colorability conjecture. We prove a fractional version of this conjecture. For this purpose, we prove a weighted generalization of a theorem of Haxell, on independent systems of representatives (ISR’s). En route, we give a survey of some recent developments in the theory of ISR’s. The research of the first author was supported by grant no 780/04 from the Israel Science Foundation, and grants from the M. & M. L. Bank Mathematics Research Fund and the fund for the promotion of research at the Technion. The research of the third author was supported by the Sacta-Rashi Foundation.     

Diagrammes de Diamond et (<Emphasis Type="Italic">φ</Emphasis>, Γ)-modules

Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /ℚ _p ) over ̅F _p . The modulo p Langlands correspondence for GL₂(ℚ _p ) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚ _p and ρ a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ _p /F) over ̅F _p . When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚ _p of the dual of ρ.     

Cyclotomic units in z p -extensions

LetK ₀ be the maximal real subfield of the field generated by thep-th root of 1 over ℚ, andK∞ be the basic Z_p-extension ofK ₀ for a fixed odd primep. LetK _n be itsn-th layer of this tower. For eachn, we denote the Sylowp-subgroup of the ideal class group ofK _n byA _n , and that ofE _n C _n byB _n , whereE _n (resp.C _n ) is the group of units (resp. cyclotomic units ofK _n . In section 2 of this paper, we describe structures of the direct and inverse limits ofB _n . The direct limit, in particular, is shown to be a direct sum of λ copies ofp-divisible groups and a finite group M, where λ is the Iwasawa λ-invariant for K∞ overK ₀. In section 3, we prove that the capitulation ofA _n inA _m is isomorphic to M form ≫n ≫ 0 by using cohomological arguments. Hence if we assume Greenberg’s conjecture (λ = 0), thenA _n is isomorphic toB _n forn ≫ 0. This paper was supported in part by a research fund for junior scholars, Korea Research Foundation The present studies were supported in part by the Basic Science Research Institute program, Ministry of Education, 1989.     

Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.     

On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.