Wojciech J. Gajda

Abstract:

It is well known that analytic sources, like zeta and L-functions, provide information on class groups. Not only the order of a class group but also its structure as a module over a group ring has been studied in this way. The strongest imaginable result would be determining class groups up to module isomorphism, but this seems too ambitious in general. A popular ``best approximation'' consists in determining the Fitting ideal. The prototypical result (we omit all hypotheses, restrictions and embellishments) predicts the Fitting ideal of a class group as the product of a certain ideal $J$ and a so-called equivariant L-value $\omega$ in a group ring. The element $\omega$ generates a principal ideal, but its definition is analytic, and rather sophisticated. On the other hand, the ideal $J$ is usually far from principal but has a much more elementary description. In this talk we intend to describe a few recent results of this type, and we explain a new concept of ``equivalence'' of modules. This leads, ideally, to a finer description of the class groups a priori than just determining their Fitting ideals. Iin other words, we are trying to improve the above-mentioned ``best approximation''. -- This is recent joint work with Takenori Kataoka.