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
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
equality is conditional on the Birch-Tate conjecture. Extensions of
this result will be described as time permits.