Iwasawa Theory of Number Fields

2008 ◽  
pp. 721-784
Author(s):  
Jürgen Neukirch ◽  
Alexander Schmidt ◽  
Kay Wingberg
Keyword(s):  
Number Fields ◽  
Iwasawa Theory

A Relation Between the 2-Primary Parts of the Main Conjecture and the Birch-Tate-Conjecture

1989 ◽  
Vol 32 (2) ◽  
pp. 248-251 ◽  
Author(s):  
Manfred Kolster
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Tate Conjecture ◽  
Main Conjecture ◽  
Totally Real

AbstractIt is shown that for totally real number fields the Main Conjecture in Iwasawa-Theory for p = 2 proposed by Fédérer implies the 2-primary part of the Birch-Tate-Conjecture in analogy with the case p odd.

scholarly journals Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

2014 ◽  
Vol 10 (08) ◽  
pp. 2045-2095 ◽  
Author(s):  
David Loeffler ◽  
Sarah Livia Zerbes
Keyword(s):  
Galois Group ◽  
Lie Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Iwasawa Theory ◽  
Absolute Galois Group ◽  
Abelian Extensions ◽  
Dimension 2 ◽  
Crystalline Representation

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

REPRESENTING Ω(l∞) FOR REAL ABELIAN FIELDS

2003 ◽  
Vol 02 (03) ◽  
pp. 237-276 ◽  
Author(s):  
JÜRGEN RITTER ◽  
ALFRED WEISS
Keyword(s):  
Real Number ◽  
Prime Number ◽  
Cyclotomic Field ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Root Number ◽  
Abelian Extensions ◽  
Main Conjecture ◽  
Abelian Fields ◽  
Totally Real

For real subfields K of a cyclotomic field ℚ(ς) we remove the tameness assumption at a given odd prime number l, which was needed in [11] in order to establish the equivalence of the Lifted Root Number Conjecture at l and an equivariant main conjecture of Iwasawa theory for abelian extensions of totally real number fields K/k.

scholarly journals Growth of in towers for isogenous curves

2015 ◽  
Vol 151 (11) ◽  
pp. 1981-2005 ◽  
Author(s):  
Tim Dokchitser ◽  
Vladimir Dokchitser
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties ◽  
Number Fields ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Growth Types

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

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.

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