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 Stickelberger elements and Kolyvagin systems

2011 ◽  
Vol 203 ◽  
pp. 123-173
Author(s):  
Kâzim Büyükboduk
Keyword(s):  
General Framework ◽  
Euler System ◽  
Real Field ◽  
Ideal Class ◽  
The Core ◽  
Ideal Class Groups ◽  
Kolyvagin Systems ◽  
Totally Real ◽  
Totally Real Fields ◽  
The Ideal

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank isr >1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

scholarly journals Stickelberger elements and Kolyvagin systems

2011 ◽  
Vol 203 ◽  
pp. 123-173
Author(s):  
Kâzim Büyükboduk
Keyword(s):  
General Framework ◽  
Euler System ◽  
Real Field ◽  
Ideal Class ◽  
The Core ◽  
Ideal Class Groups ◽  
Kolyvagin Systems ◽  
Totally Real ◽  
Totally Real Fields ◽  
The Ideal

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r > 1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

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∞).

scholarly journals On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

2013 ◽  
Vol 65 (2) ◽  
pp. 403-466
Author(s):  
Jeanine Van Order
Keyword(s):  
Modular Forms ◽  
Iwasawa Theory ◽  
The Other ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Totally Real ◽  
Totally Real Fields

AbstractWe construct a bipartite Euler systemin the sense ofHoward forHilbertmodular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies inmany cases one divisibility of the associated dihedral or anticyclotomicmain conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

scholarly journals Λ-adic modular symbols over totally real fields

2011 ◽  
pp. 841-865 ◽  
Author(s):  
Baskar Balasubramanyam ◽  
Matteo Longo
Keyword(s):  
Modular Symbols ◽  
Totally Real ◽  
Totally Real Fields
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