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 Φ be a Hayes module on the Hilbert class field H_A of F. We prove an Iwasawa Main Conjecture for the (Z_p)^∞-extension mathcal{F}/F generated by the mathfrak{p}-power torsion of Φ (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 mathfrak{p}-adic characters, interpolates the Goss Zeta-function or some mathfrak{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.