The refined Coates-Sinnott conjecture for characteristic p global fields

TL;DR

This paper proves a refined version of the Coates-Sinnott conjecture for function fields, relating K-theory Fitting ideals to special L-values using Galois module techniques inspired by Iwasawa theory.

Contribution

It provides the first proof of a refined function field analogue of the Coates-Sinnott conjecture, connecting K-theory and L-values through Galois module analysis.

Findings

Calculated Fitting ideals of specific K-groups in terms of L-values.

Extended equivariant Iwasawa theory techniques to function fields.

Linked Galois module structures to special values of L-functions.

Abstract

This article is concerned with proving a refined function field analogue of the Coates-Sinnott conjecture, formulated in the number field context in 1974. Our main theorem calculates the Fitting ideal of a certain even Quillen K-group in terms of special values of L-functions. The techniques employed are directly inspired by recent work of Greither and Popescu in the equivariant Iwasawa theory of arbitrary global fields. They rest on the results of Greither-Popescu on the Galois module structure of certain naturally defined Picard 1-motives associated to an arbitrary Galois extension of function fields.