Unit L-functions for \'etale sheaves of modules over noncommutative rings

TL;DR

This paper develops a theory of unit L-functions for étale sheaves of modules over noncommutative rings, establishing a key algebraic property and applying it to prove a noncommutative Iwasawa main conjecture for p-adic Lie coverings.

Contribution

It introduces a canonical preimage of certain L-functions in noncommutative K-theory and applies this to prove a noncommutative Iwasawa main conjecture in algebraic geometry.

Findings

Established a canonical preimage of L-functions in K_1( ext{Lambda}[T])

Proved a version of the noncommutative Iwasawa main conjecture

Extended L-function theory to noncommutative sheaves over finite fields

Abstract

Let $s\colon X\rightarrow \operatorname{Spec} \mathbb{F}$ be a separated scheme of finite type over a finite field $\mathbb{F}$ of characteristic $p$, let $\Lambda$ be a not necessarily commutative $\mathbb{Z}_p$-algebra with finitely many elements, and let $\mathcal{F}^\bullet$ be a perfect complex of $\Lambda$-sheaves on the \'etale site of $X$. We show that the ratio $L(\mathcal{F}^\bullet,T)/L(R s_!\mathcal{F}^\bullet,T)$, which is a priori an element of $K_1(\Lambda[[T]])$, has a canonical preimage in $K_1(\Lambda[T])$. We use this to prove a version of the noncommmutative Iwasawa main conjecture for $p$-adic Lie coverings of $X$.