The $l$-adic $K$-theory of a $p$-local field

TL;DR

This paper verifies a special case of Carlsson's conjecture relating the $ extit{l}$-adic K-theory of a field to Galois group representations, focusing on the Laurent series over a finite field.

Contribution

It confirms Carlsson's conjecture for the case of Laurent series fields over finite fields, expanding the known instances where the conjecture holds.

Findings

Verification of Carlsson's conjecture for $ extit{l}$-adic K-theory of $f_{p}((x))$

Establishment of the relationship between K-theory and Galois representations in this case

Extension of the conjecture's validity to new field classes

Abstract

We verify a special case of a conjecture of G. Carlsson that describes the $\l$-adic $K$-theory of a field $F$ of characteristic prime to $\l$ in terms of the representation theory of the absolute Galois group $G_F$. This conjecture is known to hold in two cases; in this article we examine the second case, in which the field in question is the field of Laurent series over a finite field $\FF_{p}((x))$.