Iwasawa theory of Rubin-Stark units and class groups

2016 ◽  
Vol 153 (3-4) ◽  
pp. 403-430 ◽  
Author(s):  
Youness Mazigh
Keyword(s):  
Iwasawa Theory ◽  
Class Groups ◽  
Stark Units

scholarly journals ON IWASAWA THEORY OF RUBIN–STARK UNITS AND NARROW CLASS GROUPS

2018 ◽  
Vol 61 (03) ◽  
pp. 673-691
Author(s):  
YOUNESS MAZIGH
Keyword(s):  
Finite Extension ◽  
Projective Limit ◽  
Iwasawa Theory ◽  
Class Groups ◽  
Exterior Power ◽  
Totally Positive ◽  
Stark Units ◽  
Totally Real ◽  
Characteristic Ideal ◽  
Narrow Class

AbstractLet K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters χ of Gal(L∞/K∞).

Théorie d’Iwasawa des unités de Stark et groupe de classes

2017 ◽  
Vol 13 (05) ◽  
pp. 1165-1190 ◽  
Author(s):  
Jilali Assim ◽  
Youness Mazigh ◽  
Hassan Oukhaba
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Projective Limit ◽  
Ideal Class ◽  
Class Groups ◽  
Euler Systems ◽  
Ideal Class Groups ◽  
Stark Units ◽  
Characteristic Ideal ◽  
The Ideal

Let [Formula: see text] be a number field and let [Formula: see text] be an odd rational prime. Let [Formula: see text] be a [Formula: see text]-extension of [Formula: see text] and let [Formula: see text] be a finite extension of [Formula: see text], abelian over [Formula: see text]. In this paper we extend the classical results, e.g. [16], relating characteristic ideal of the [Formula: see text]-quotient of the projective limit of the ideal class groups to the [Formula: see text]-quotient of the projective limit of units modulo Stark units, in the non-semi-simple case, for some [Formula: see text]-irreductible characters [Formula: see text] of [Formula: see text]. The proof essentially uses the theory of Euler systems.

scholarly journals Stark units and the main conjectures for totally real fields

2009 ◽  
Vol 145 (5) ◽  
pp. 1163-1195 ◽  
Author(s):  
Kâzım Büyükboduk
Keyword(s):  
System Theory ◽  
Iwasawa Theory ◽  
The Core ◽  
Kolyvagin Systems ◽  
Leopoldt's Conjecture ◽  
Stark Units ◽  
Stark Conjecture ◽  
Totally Real ◽  
Totally Real Fields ◽  
Iwasawa Algebra

AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

scholarly journals An extension theorem for integral representations

1997 ◽  
Vol 63 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Finite Group ◽  
Matrix Representation ◽  
Extension Theorem ◽  
Iwasawa Theory ◽  
Integral Representations ◽  
Roots Of Unity ◽  
Cyclotomic Fields ◽  
Qualitative Result ◽  
Class Groups ◽  
A Finite Group

AbstractBy a fundamental theorem of Brauer every irreducible character of a finite group G can be written in the field Q(εm) of mth roots of unity where m is the exponent of G. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

On p-adic L-series, p-adic cohomology and class field theory

2017 ◽  
Vol 2017 (732) ◽  
pp. 55-83 ◽  
Author(s):  
David Burns ◽  
Daniel Macias Castillo
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Iwasawa Theory ◽  
Class Groups ◽  
Class Field ◽  
Weil Theorem ◽  
Positive Integers ◽  
Natural Families

Abstract We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

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