Rubin-Stark elements and ideal class groups (Algebraic Number Theory and Related Topics 2013)

Abstract

This article is a survey of the Galois module structure of the class groups of number fields, and of their relation with the L-values. After we explain several classical results, e.g., the order of a character component of a class group, etc., we introduce the Rubin-Stark conjecture in Rubin [27], which asserts the existence of certain algebraic elements related to the L-values. We also explain some new properties of Rubin-Stark elements obtained in a joint work of the author with D. Burns and T. Sano [7], including the description of the Fitting ideals of class groups and certain cohomology groups. We also give several concrete examples.