On Iwasawa Theory over Function Fields

TL;DR

This paper proves that the characteristic ideal of the $p$-completion of the class group in a $ ext{Z}_p^d$-extension over a function field is generated by a Stickelberger element, linking class groups and special values of $L$-functions.

Contribution

It establishes a function field analogue of Iwasawa theory by connecting class groups with Stickelberger elements in a new setting.

Findings

Characteristic ideal generated by Stickelberger element

Connection between class groups and $L$-function values

Extension of Iwasawa theory to function fields

Abstract

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module $\Cl_{k_{\infty},p}$ is generated by the Stickelberger element of Gross which calculates the special values of $L$ functions.