Tate sequences and lower bounds for ranks of class groups

Cornelius Greither¹

¹ Fakultät Informatik Universität der Bundeswehr München 85577 Neubiberg, Germany

Acta Arithmetica, Tome 160 (2013) no. 1, pp. 55-66.

Résumé

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.

Détail

DOI : 10.4064/aa160-1-4

Keywords: tate sequences play major role modern algebraic number theory extension class tate sequence subtle invariant which comes class field theory hard grasp short paper demonstrate extract information tate sequence without knowing extension class particular situations certain totally real fields lower bounds rank ell part class group certain fields lower bounds minus part ell part class group these results reprove partly generalise earlier results cornell rosen era author methods purely algebraic involving little cohomology