C. Greither - On Brumer's conjecture

Stickelberger showed around 1890 that a certain ideal I in the integral group ring annihilates the class group of an abelian extension L with Galois group G over the field of rational numbers. The ideal I is constructed algebraically, but it has an interpretation by means of special values of L-functions as well. Stickelberger's proof is constructive in the sense that the required generators of principal ideals are written down as explicitly as one could ask for.

If now L is a complex abelian extension of a totally real field K, one can construct a Brumer-Stickelberger ideal I by reversing history, that is, by starting from special L-values. The Brumer conjecture then predicts annihilation of the class group Cl(K) by I (with some precautions); it is not proved yet in general. The talk will explain how to prove this conjecture for some classes of extensions L/K. The essential ingredients are the theory of Fitting ideals, Iwasawa theory, and the clever method of Wiles to bypass trivial zeros. Nobody knows how to get the required generators explicitly; the proof is highly nonconstructive.

At the end of the talk, some numerical and heuristical remarks concerning the -structure of class groups will be presented, in the spirit of the Cohen-Lenstra heuristics.