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Iwasawa theory of p-adic Lie extensions

Venjakob, Otmar

German Title: Iwasawa-Theorie p-adischer Lie-Erweiterungen

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Translation of abstract (English)

The Iwasawa theory of p-adic Lie groups investigates arithmetic objects above infinite field extensions of a number field k whose Galois group is a p-adic analytic group. The most prominent example (due to Serre) is produced by adjoining the p-torsion points of an elliptic curve defined over k without complex multiplication. The strategy consists in considering the Selmer Group or other cohomology groups which 'live' above the p-adic Lie extension as a module over the (non-commutative) group algebra R of G with coefficients in the p-adic integers. In the first, algebraic part of this dissertation special properties of R and of finitely generated R-modules are studied. In particular, we introduce the notation of pseudo-null modules as well as pseudo-isomorphisms, which turn out to be essential for structure theorems of R-modules. Then a local duality theorem and the Auslander-Buchsbaum equality for R are proved. In the second, arithmetic part we show the existence of certain pseudo-isomorphisms of global Iwasawa modules, we study the µ-invariant and we prove for some Galois modules that they do not contain any non-trivial pseudo-null submodules.

Document type: Dissertation
Supervisor: Wingberg, Prof. Dr. Kay
Date of thesis defense: 14 March 2001
Date Deposited: 28 Mar 2001 00:00
Date: 2001
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
DDC-classification: 510 Mathematics
Controlled Keywords: Iwasawa-Theorie, Galois-Kohomologie, Lokale Kohomologie, Elliptische Kurve
Uncontrolled Keywords: Abelsche Varietät , Auslander reguläre Ringe , Auslander-Buchsbaum GleichungAbelean variety , Auslander regular ring , Galois-Cohomology , Auslander-Buchsbaum equality
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