Coleman-adapted Rubin–Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties

2015 ◽  
Vol 111 (6) ◽  
pp. 1338-1378 ◽  
Author(s):  
Kâzım Büyükboduk ◽  
Antonio Lei
Keyword(s):  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Kolyvagin Systems

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 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 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.

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