On the Serre conjecture for Artin characters in the geometric case

TL;DR

This paper proves Serre's conjecture that a higher-dimensional analogue of the Artin character, defined via group actions on regular local rings, corresponds to a rational representation in the geometric setting.

Contribution

It establishes the conjecture for the case of equal characteristic, linking the higher-dimensional Artin-like function to rational representations.

Findings

Proves Serre's conjecture in the equal characteristic case.

Shows the function a_G corresponds to a rational representation.

Extends the understanding of Artin characters in geometric contexts.

Abstract

Let $G$ be a finite group and $A$ be a regular local ring on which $G$ acts. Under certain assumptions on $A$ and the action, Serre defined a function $a_G\colon G\rightarrow\mathbb{Z}$ which can be viewed as a higher dimensional analogue of Artin character, and conjectured that it is associated to a $\mathbb{Q}_\ell$-rational representation of $G$ for any prime $\ell$ invertible in $A$. We prove this conjecture in the equal characteristic case.