Stickelberger series and Main Conjecture for function fields

TL;DR

This paper proves an Iwasawa Main Conjecture for function fields using a Stickelberger series that links algebraic class groups with analytic L-functions, advancing understanding in arithmetic geometry.

Contribution

It establishes the Iwasawa Main Conjecture for certain function field extensions and introduces a Stickelberger series that interpolates Goss Zeta-functions and p-adic L-functions.

Findings

Proved the Iwasawa Main Conjecture for the specified extension.

Constructed a Stickelberger series that generates the Fitting ideal of the class group.

Demonstrated the series interpolates both Goss Zeta-functions and p-adic L-functions.

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.