Noncommutative Main Conjectures of Geometric Iwasawa Theory

TL;DR

This paper surveys noncommutative main conjectures in geometric Iwasawa theory for schemes over finite fields, highlighting key results and discussing related conjectures for function fields.

Contribution

It provides a comprehensive overview of the state of noncommutative Iwasawa main conjectures in a geometric context, including recent proofs and conjectural extensions.

Findings

Burns and the author proved key cases of the conjecture.

The survey summarizes known results and open problems.

Brief discussion on conjectures for function fields.

Abstract

We give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for separated schemes of finite type over a finite field, as stated and proved by Burns and the author. We will also comment briefly on versions of the main conjecture for function fields.