On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space Ώ _L ^s (0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module Ώ _L ^s (0) into a direct sum of indecomposablek[G]-modules. Partially supported by CONACyT, project No. 25063-E.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Ramification de certaines extensions de Lie p-adiques

Let G be a p-adic Lie group and let K be a finite extension of the p-adic number field ℚ _p . There are finitely many filtrations of G which could be ramification filtrations of totally ramified Galois extensions of K with Galois group G. Received: 19 October 1998     

On the class numbers of certain Hilbert class fields

LetL/k be a finite Galois extension with Galois groupG, and a group extension. We study the existence of the Galois extensionM/L/k such that the canonical projection Gal(M/k)→Gal(L/k) coincides with the given homomorphismj:E→G and thatM/L is unramified.     

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).     

A note on a theorem of W. Gaschütz and N. Itô

Given a finite groupG andp an odd prime number, we conclude thatO ^p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK.     

Galois groups ofp-extensions with two ramification points

LetK _p (p, q) be the maximalp-extension of the field ℚ of rational numbers with ramification pointsp andq. LetG _p (p, q) be the Galois group of the extensionK _p(p.q)/ℚ. It is known thatG _p(p, q) can be presented by two generators which satisfy a single relation. The form of this relation is known only modulo the second member of the descending central series ofG _p(p, q). In this paper, we find an arithmetical-type condition on which the form of the relation modulo the third member of the descending central series ofG _p(p, q) depends. We also consider two examples withp=3,q=19 andp=3,q=37. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 48–60, January–March, 2000. Translated by H. Markšaitis     

Sharp bounds for the number of 3‐independent partitions and the chromaticity of bipartite graphs

Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ??^?s(p,q) (resp., ??₂^?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph K_p,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ^?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? ₂^?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ^?s(p,q) (resp., G ∈ ?? ₂^?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ??^?s_i(p+i,q?i) where max and s_i = s ? i(p?q+i) (resp., a subset of ??₂^?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from K_p,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001     

Covers of varieties with completely split points

LetK be a field such that all Sylow subgroups of its absolute Galois groupG _Kare infinite. LetX be a smooth variety overK with function fieldF andY→X the normalisation in a finite, separable extensionE/vbF. We show: If there is a closed pointx∈X which does not split completely inY→X, then the set of these points is Zariski dense inX.     

Galois Structure of K-Groups of Rings of Integers

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts, to the root number class of Fröhlich and Cassou-Noguès.     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

Construction of somep-extensions of the rational number field

In this paper, we constructp-extensionsK _a ,a(modp ^r ), of degreep ^3r,p≠2, r>0, of the field ℚ of rational numbers with ramification pointsp andq. The Galois groupG(K _a )/ℚ of the extensionK _a /ℚ,a(modp ^r ), is defined by the generators and relations

NILPOTENT GROUPS AND ABELIANIZATION

We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having q^th roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups.

Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [H_P, K_p]

Let the group G act on the nilpotent group N. Then G acts compatibly on N_p and (Γ^G _i N)_P = Γ^G _i(N_p).

The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4].     

Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups

For a locally compact groupG and a groupB of topological automorphisms containing the inner automorphisms ofG and being relatively compact with respect to Birkhoff topology (that isG[FIA] _B,B I(G)) the spaceG _B of -orbits is a commutative hypergroup (=commutative convo inJewett's terminology) in a natural way asJewett has shown. Identifying the space of hypergroup characters ofG _B withE(G, B) (the extreme points ofB-invariant positive definite continuous functionsp withp (e)=1, endowed with the topology of compact convergence) we prove thatE(G, B) is a hypergroup, the hypergroup dual ofG _B.     

On products of additive functions (a third approach)

Summary Leta ₁, , a_s : G K be additive functions from an abelian groupG into a fieldK such thata ₁(g)··a_s(g) = 0 for allg G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.     

K-theory for finite covariant systems

LetA be a real or complex Banach algebra and assume that is an action of a finite groupG onA by means of continuous automorphisms. To such a finite covariant system (A, G, ), we associate an Abelian groupK(A, G, ). We obtain some classical exact sequences for an algebraA and a closed invariant idealI. We also compute the group in a few important special cases. Doing so, we relate our new invariant to the classicalK ₀ andK ₁ of a Banach algebra and to theK-theory of ₂-graded Banach algebras. Finally, we obtain a result that gives a close relationship of our groupK(A, G, ) with theK-theory of the crossed productA G. In particular, we prove a six-term exact sequence involving our groupK(A, G, ) and theK-groups ofA G. In this way, we hope to contribute to the well-known problem of finding theK-theory of the crossed productA G in the case of an action of a finite group.     

The Langlands classification for non-connectedp-adic groups

We give the Langlands classification for a non-connected reductive quasisplitp-adic groupG, under the assumption thatG/G ⁰ is abelian (here,G ⁰ denotes the connected component of the identity ofG). The Langlands classification for non-connected groups is an extension of the Langlands classification from the connected case.     

Ki-covers. II. Ki-perfect graphs

AK_i is a complete subgraph of size i. A K_i-cover of a graph G(V, E) is a set C of K_i?1s of G such that every K_i in G contains at least one K_i?1 in C . c_i(G) is the cardinality of a smallest K_i-cover of G. A K_i-packing of G is a set of K_is such that no two K_is have i ? 1 nodes in common. p_i(G) is the cardinality of a largest K_i-packing of G. Let F _i(G) denote the set of K_is in G and define c_i(F) and p_i(F) analogously for F ? F _i(G). G is K_i-perfect if ?F ? F _i(G), c_i(F) = p_i(F). The K₂-perfect graphs are precisely the bipartite graphs. We present a characterization of K_i-perfect graphs that is similar to the Strong Perfect Graph Conjecture, and explore the relationships between K_i-perfect graphs and normal hypergraphs. Furthermore, if iA denotes the 0 ? 1 matrix of G where the rows are the elements of F _i?1(G) that belong to at least one K_i and the columns are the elements of F _i(G), then we show that iA is perfect iff G is a K_i-perfect graph. We also characterize the K_i-perfect graphs for which iAis balanced.     

Generalized frobenius groups. II

A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C _G (x)|=|C _G/K (xK)| for allx∈G\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.