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Tate sequences and lower bounds for ranks of class groups

Cornelius Greither — 2013

Acta Arithmetica

Tate sequences play a major role in modern algebraic number theory. The extension class of a Tate sequence is a very subtle invariant which comes from class field theory and is hard to grasp. In this short paper we demonstrate that one can extract information from a Tate sequence without knowing the extension class in two particular situations. For certain totally real fields K we will find lower bounds for the rank of the ℓ-part of the class group Cl(K), and for certain CM fields we will find lower...

Class groups of abelian fields, and the main conjecture

Cornelius Greither — 1992

Annales de l'institut Fourier

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture is stated...

Non-existence and splitting theorems for normal integral bases

Cornelius GreitherHenri Johnston — 2012

Annales de l’institut Fourier

We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower K L forces the tower to be split in a very strong sense.

The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems

Cornelius GreitherRadiu Kučera — 2002

Annales de l’institut Fourier

The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s Ω ( 3 ) - conjecture for Galois extensions K / F of number fields. It is certainly more difficult than the Ω ( 3 ) -localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that F = and the degree of K / F is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important...

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