Stickelberger series and Main Conjecture for function fields

2021/03/17 by Bandini, Andrea, Coscelli, Edoardo
#11M38 #11R23 #11R58 #11R60 #11S40 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2103.09667

Abstract

Let F be a global function field of characteristic p with ring of integers A and let Φbe a Hayes module on the Hilbert class field H(A) of F. We prove an Iwasawa Main Conjecture for the Z_p^∞-extension F/F generated by the \mathfrakp-power torsion of Φ(\mathfrakp 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.

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