On the Iwasawa theory of p-adic Lie extensions

TL;DR

This paper advances the understanding of Iwasawa modules over p-adic Lie extensions by applying new non-commutative algebra techniques to relate Selmer groups, Galois groups, and local units in number theory.

Contribution

It introduces novel methods for analyzing modules over non-commutative Iwasawa algebras and explores their arithmetic implications in p-adic Lie extensions.

Findings

Established a relationship between Selmer groups and Galois groups in p-adic Lie extensions.

Determined the Galois module structure of local units and related modules.

Applied new techniques to the structure theory of modules over non-commutative Iwasawa algebras.

Abstract

In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism". In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.