STICKELBERGER SERIES AND MAIN CONJECTURE FOR FUNCTION FIELDS

Let F be a global function field of characteristic p with ring of integers A and let Phi be a Hayes module on the Hilbert class field H-A of F. We prove an Iwasawa Main Conjecture for the Z(p)(infinity)-extension F/F generated by the p-power torsion of Phi (p a prime of A). The main tool is a Stickelberger series whose specialization provides a generator for the Fitting ideal of the class group of F. Moreover we prove that the same series, evaluated at complex or p-adic characters, interpolates the Goss Zeta-function or some p-adic L-function, thus providing the link between the algebraic structure (class groups) and the analytic functions, which is the crucial part of Iwasawa Main Conjecture.