Geometric Main Conjectures in Function Fields

TL;DR

This paper proves an Equivariant Main Conjecture in Iwasawa Theory for function fields, extending classical results to a broader setting involving Drinfeld modules and Iwasawa modules.

Contribution

It establishes the conjecture in a new context of function fields with Drinfeld modules, generalizing prior cyclotomic Iwasawa theory results.

Findings

Proves the Equivariant Main Conjecture for a class of function fields.

Extends classical Iwasawa theory results to Drinfeld modules.

Connects Iwasawa modules with recent work by Greither and Popescu.

Abstract

We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by Greither and Popescu, in their proof of the classical Equivariant Main Conjecture along the (arithmetic) cyclotomic Iwasawa tower.