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Signed Selmer groups over p -adic Lie extensions

Antonio Lei, Sarah Livia Zerbes (2012)

Journal de Théorie des Nombres de Bordeaux

Let E be an elliptic curve over ℚ with good supersingular reduction at a prime p ≥ 3 and a p = 0 . We generalise the definition of Kobayashi’s plus/minus Selmer groups over ℚ ( μ p ∞ ) to p -adic Lie extensions K ∞ of ℚ containing ℚ ( μ p ∞ ) , using the theory of ( ϕ , Γ ) -modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case....

Stabilization in non-abelian Iwasawa theory

Andrea Bandini, Fabio Caldarola (2015)

Acta Arithmetica

Let K/k be a ℤₚ-extension of a number field k, and denote by kₙ its layers. We prove some stabilization properties for the orders and the p-ranks of the higher Iwasawa modules arising from the lower central series of the Galois group of the maximal unramified pro-p-extension of K (resp. of the kₙ).

Structure of central torsion Iwasawa modules

Susan Howson (2002)

Bulletin de la Société Mathématique de France

We describe an approach to determining, up to pseudoisomorphism, the structure of a central-torsion module over the Iwasawa algebra of a pro- p , p -adic, Lie group containing no element of order p . The techniques employed follow classical methods used in the commutative case, but using Ore’s method of localisation. We then consider the properties of certain invariants which may prove useful in determining the structure of a module. Finally, we describe the case of pro- p subgroups of GL 2 ( ℤ p ) in detail and...

Sur la dimension cohomologique des pro- p -extensions des corps de nombres

Christian Maire (2005)

Journal de Théorie des Nombres de Bordeaux

In this paper, we study the cohomological dimension of groups G S : = Gal ( 𝕂 S / 𝕂 ) , where 𝕂 S is the maximal pro- p -extension of a number field 𝕂 , unramified outside a finite set S of places of 𝕂 . This dimension is well-understood only when S contains all places above p ; in the case where only some of the places above p are contained in S , one can still obtain some results if 𝕂 S / 𝕂 contains at least one ℤ p -extension 𝕂 ∞ / 𝕂 . Indeed, in that case, the study of the ℤ p [ [ Gal ( 𝕂 ∞ / 𝕂 ) ] ] -module Gal ( 𝕂 S / 𝕂 ∞ ) a b allows one to give sufficient conditions for the pro- p -group...

Sur la dualité et la descente d’Iwasawa

David Vauclair (2009)

Annales de l’institut Fourier

Nous développons – en nous appuyant sur l’exemple concret des unités cyclotomiques et du groupe de classes en théorie d’Iwasawa cyclotomique – de nouveaux outils pour une étude générale de la descente et de la codescente, dans l’optique de comparer ces deux points de vue duaux.Si A = ( A n ) est un « système normique » (i.e. une collection de modules galoisiens avec données supplémentaires), attaché à une extension de Lie p -adique fixée d’algèbre d’Iwasawa Λ , nous montrons principalement qu’il existe un...

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