Stickelberger series and Main Conjecture for function fields

Abstract

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^\infty-extension F/F generated by the \mathfrak{p}-power torsion of \Phi (\mathfrak{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.