Artin L-function on PGL3

TL;DR

This paper investigates Artin L-functions associated with PGL3 on a 2-dimensional complex, establishing key properties like rationality and functional equations, and connecting them to the L-functions of representations via cohomological methods.

Contribution

It extends the understanding of Artin L-functions on PGL3, proving fundamental properties and linking them to representation L-functions through cohomology.

Findings

Proved rationality and functional equations of Artin L-functions.

Established invariance under induction for these L-functions.

Connected Artin L-functions to representation L-functions via cohomological methods.

Abstract

We study Artin $L$-functions on a finite $2$-dimensional complex $X_\Gamma$ arising from PGL$_3$ attached to finite-dimensional representations $\rho$ of its fundamental group. Some key properties, such as rationality, functional equation, and invariance under induction, of these functions are proved. Moreover, using a cohomological argument, we establish a connection between the Artin $L$- functions and the $L$-function of $\rho$, extending the identity on zeta functions of $X_\Gamma$ obtained in [KL,KLW].