Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

TL;DR

This paper proves an Iwasawa Main Conjecture for a function field cyclotomic extension, linking class groups, Stickelberger elements, and p-adic L-functions, and derives results on Bernoulli-Goss numbers and zeta-values.

Contribution

It establishes the Iwasawa Main Conjecture for the class group of a function field cyclotomic extension, connecting it with Stickelberger elements and p-adic L-functions.

Findings

Fitting ideal generated by a Stickelberger element

Analog of Ferrero-Washington theorem for the function field extension

Information on p-adic valuations of Bernoulli-Goss numbers

Abstract

We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(\theta)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[\theta]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss $\zeta$-function at negative integers).