Stickelberger Elements for Cyclic Extensions and the Order of Vanishing of Abelian L-Functions at s=0
We study the Stickelberger element of a cyclic extension of global fields of prime power degree. Assuming that S contains an almost splitting place, we show that the Stickelberger element is contained in a power of the relative augmentation ideal whose exponent is at least as large as Gross's prediction. This generalizes the work of Tate (see Section 4) on a refinement of Gross's conjecture in the cyclic case. We also present an example for which Tate's prediction does not hold.
Key Words: class numbers; Gross's conjecture; special values of Abelian L-functions; Stickelberger elements.