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 bounds for the minus part of the ℓ-part of the class group. These results reprove and partly generalise earlier results by Cornell and Rosen, and by R. Kučera and the author. The methods are purely algebraic, involving a little cohomology.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-4,
author = {Cornelius Greither},
title = {Tate sequences and lower bounds for ranks of class groups},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {55-66},
zbl = {1284.11150},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-4}
}
Cornelius Greither. Tate sequences and lower bounds for ranks of class groups. Acta Arithmetica, Tome 161 (2013) pp. 55-66. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-4/