Jonathan Sands, University of Vermont

"Some Galois module structure analogies between tame kernels and ideal class groups"

Abstract: Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of 
number fields E/F should annihilate the ideal class group of E, when it is considered as a module over the appropriate group ring.
We describe some cases in which an even stronger relationship is known to hold: an ideal obtained from these Stickelberger elements equals
a Fitting ideal connected with the ideal class group.  We consider the analog of this at s=-1, where the class group
is replaced by the tame kernel, which we will define.  Our recent work shows that for a field extension of degree 2, there is 
an exact equality between the Fitting ideal of an enlarged tame kernel and the most natural higher Stickelberger ideal; the 2-part of this 
equality is conditional on the Birch-Tate conjecture. Extensions of this result will be described as time permits.