
C. GREITHER, Département de mathématiques et statistique, Université de Laval, Montréal, Québec G1K 7P4, Canada 
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 Lfunctions 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 BrumerStickelberger ideal I by reversing history, that is, by starting from special Lvalues. 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 CohenLenstra heuristics.