scholarly journals Refined Iwasawa theory and Kolyvagin systems of Gauss sum type

2011 ◽  
Vol 104 (4) ◽  
pp. 728-769 ◽  
Author(s):  
Masato Kurihara
Keyword(s):  
Iwasawa Theory ◽  
Gauss Sum ◽  
Kolyvagin 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 Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

2019 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Euler System ◽  
Base Change ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Imaginary Quadratic Fields ◽  
Elliptic Modular Form

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

scholarly journals Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

2018 ◽  
Vol 30 (4) ◽  
pp. 887-913 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Main Conjecture ◽  
Elliptic Modular Form

Abstract This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic {\mathbb{Z}_{p}} -tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

Iwasawa theory for motives

1991 ◽  
pp. 211-234 ◽  
Author(s):  
Ralph Greenberg
Keyword(s):  
Iwasawa Theory
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