On the hermitian structure of Galois modules in number fields and Adams operations

THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

International Journal of Number Theory ◽

10.1142/s1793042111004010 ◽

2011 ◽

Vol 07 (01) ◽

pp. 87-99 ◽

Cited By ~ 1

Author(s):

PAUL BUCKINGHAM

Keyword(s):

Class Group ◽

Arbitrary Order ◽

Number Fields ◽

Simple Zero ◽

Abelian Extension ◽

Class Groups ◽

Original Proof ◽

Galois Modules ◽

Fitting Ideal ◽

Stickelberger Ideal

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

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Self-duality and torsion Galois modules in number fields

Journal of Algebra ◽

10.1016/0021-8693(84)90210-2 ◽

1984 ◽

Vol 90 (1) ◽

pp. 230-246 ◽

Cited By ~ 1

Author(s):

Maryse Desrochers

Keyword(s):

Number Fields ◽

Self Duality ◽

Galois Modules

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FACTOR EQUIVALENCE OF GALOIS MODULES AND REGULATOR CONSTANTS

International Journal of Number Theory ◽

10.1142/s1793042113500772 ◽

2014 ◽

Vol 10 (01) ◽

pp. 1-12

Author(s):

ALEX BARTEL

Keyword(s):

Number Fields ◽

The Other ◽

Module Structure ◽

Integral Group ◽

Galois Module ◽

Group Representations ◽

Galois Modules ◽

Rings Of Integers ◽

The One ◽

Module Structures

We compare two approaches to the study of Galois module structures: on the one hand, factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand, regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We then use this comparison to derive a factorizability result on higher K-groups of rings of integers, which is a direct analogue of a theorem of de Smit on S-units.

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ON THE GALOIS STRUCTURE OF ARITHMETIC COHOMOLOGY I: COMPACTLY SUPPORTED -ADIC COHOMOLOGY

Nagoya Mathematical Journal ◽

10.1017/nmj.2018.41 ◽

2018 ◽

Vol 239 ◽

pp. 294-321

Author(s):

DAVID BURNS

Keyword(s):

Lower Bounds ◽

Projective Module ◽

Number Fields ◽

Upper And Lower Bounds ◽

Selmer Groups ◽

Cohomology Groups ◽

Field Extensions ◽

Direct Sum ◽

Galois Modules ◽

Galois Structure

We investigate the Galois structures of $p$-adic cohomology groups of general $p$-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded $p$-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.

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The isomorphism class of a set of lattices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100067682 ◽

1989 ◽

Vol 105 (2) ◽

pp. 197-210 ◽

Cited By ~ 1

Author(s):

S. M. J. Wilson

Keyword(s):

Algebraic Number ◽

Isomorphism Class ◽

Number Fields ◽

Algebraic Number Fields ◽

Ring Of Integers ◽

Isomorphism Classes ◽

Finite Dimensional ◽

Galois Modules ◽

Stable Class ◽

Residue Fields

AbstractLet R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

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