Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications

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).

arXiv commentSection 3 entirely rewritten