Stickelberger ideals and divisor class numbers

Abstract

Let K/k be a finite abelian extension of function fields with Galois group G. Using the Stickelberger elements associated to K/k studied by J. Tate, P. Deligne and D. Hayes, we construct an ideal I in the integral group ring $\mathbb{Z}[G]$ relative to the extension K/k, whose elements annihilate the group of divisor classes of degree zero of K and whose rank is equal to the degree of the extension. When K/k is a (wide or narrow) ray class extension, we compute the index of I in $\mathbb{Z}[G]$ , which is equal to the divisor class number of K up to a trivial factor.