Stickelberger ideals and divisor class numbers (Q1601776)

\textit{K. Iwasawa} introduced in [Ann. Math. (2) { 76}, 171--179 (1962; Zbl 0125.02003)], using the Stickelberger elements, the notion of Stickelberger ideal whose elements annihilate the ideal class group of the cyclotomic field of prime power conductor. Iwasawa showed that the index of the Stickelberger ideal in the minus part of the group ring is equal to the relative class number of the cyclotomic field. In [Ann. Math. (2) {108}, 107--134 (1978; Zbl 0395.12014)] \textit{W. Sinnott} extends the results of Iwasawa to a cyclotomic field by introducing cohomology to the computation of the indices. In the function field case, by Weil's theorem, partial zeta functions are rational functions of \(q ^ {-s}\) where \(q\) denotes the cardinality of the field of constants. This makes it possible to construct a larger Stickelberger ideal compared to the number field case. In the paper under review the author gives a new definition of the Stickelberger ideal. For a finite abelian extension \(K/k\) of global function fields with Galois group \(G\), using the Stickelberger elements associated to \(K/k\), studied by Tate, Deligne and Hayes, an ideal is defined, called the Stickelberger ideal of \(K\), in the integral group ring \({\mathbb Z}[G]\). The elements of the ideal annihilate the group of divisor classes of degree zero of \(K\) and the rank of the ideal is equal to the degree of the extension \(K/k\). When \(K/k\) is a (wide or narrow) ray class extension of function fields the index of the Stickelberger ideal in the integral group ring is equal to the divisor class number of \(K\) up to an explicit factor.