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 $\mathbb{Z}_p^\infty$-extension $\mathcal{F}/F$ generated by the $\mathfrak{p}$-powertorsion of $\Phi$ ($\mathfrak{p}$ a prime of $A$). The main tool is a Stickelberger series whose specializationprovides a generator for the Fitting ideal of the class group of $\mathcal{F}$. 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.