Iwasawa-theory of abelian varieties at primes of non-ordinary reduction

1995 ◽  
Vol 87 (1) ◽  
pp. 225-258 ◽  
Author(s):  
Heiko Knospe
Keyword(s):  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Ordinary Reduction

scholarly journals On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

scholarly journals On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

2019 ◽  
Author(s):  
Chan-Ho Kim ◽  
Masato Kurihara
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Error Term ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Main Conjecture ◽  
Finite Layer ◽  
Ordinary Reduction ◽  
Explicit Comparison

AbstractIn this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the “weak main conjecture” à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the “strong main conjecture” suggested by the second named author under the validity of the $\pm $-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among “finite layer objects”, “$\pm $-objects”, and “fine objects” in Iwasawa theory. The case of good ordinary reduction is also treated.

Bounding the Iwasawa invariants of Selmer groups

2020 ◽  
pp. 1-33
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Analogous Result ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Selmer Groups ◽  
Iwasawa Invariants ◽  
Ordinary Reduction

Abstract We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Characteristic elements, pairings and functional equations over the false Tate curve extension

2008 ◽  
Vol 144 (3) ◽  
pp. 535-574 ◽  
Author(s):  
GERGELY ZÁBRÁDI
Keyword(s):  
Functional Equation ◽  
Iwasawa Theory ◽  
Selmer Group ◽  
Ground Field ◽  
Principal Ideals ◽  
Cyclotomic Extension ◽  
Curve Extension ◽  
Ordinary Reduction ◽  
Tate Curve ◽  
Rule Out

AbstractWe construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a primep≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of thep-adicL-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

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