Tate sequences and lower bounds for ranks of class groups
Volume 160 / 2013
Abstract
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 $\ell $-part of the class group ${\rm Cl}(K)$, and for certain CM fields we will find lower bounds for the minus part of the $\ell $-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.
