On a Noncommutative Iwasawa Main Conjecture for Function Fields

TL;DR

This paper formulates and proves a non-commutative Iwasawa Main Conjecture for function fields, establishing a functional equation for non-commutative L-functions and extending key conjectures to new algebraic contexts.

Contribution

It introduces a non-commutative Iwasawa Main Conjecture for function fields and proves a functional equation for associated L-functions, extending classical conjectures.

Findings

Proved the non-commutative Iwasawa Main Conjecture for function fields.

Established a functional equation for non-commutative L-functions.

Generalized the main conjecture to Picard-1-motives and abelian varieties.

Abstract

We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\ell$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a functional equation for the resulting non-commutative $L$-functions. As corollaries, we obtain non-commutative generalisations of the main conjecture for Picard-$1$-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the $\operatorname{GL}_2$ main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.