Twisted Poincare Series and Zeta functions on finite quotients of buildings

TL;DR

This paper generalizes the relationship between Ihara zeta functions and twisted Poincaré series from SL2 over local fields to other rank-two groups, proposing a conjecture for higher ranks.

Contribution

It extends known results for SL2 to other split simple algebraic groups of rank two and formulates a conjecture for higher rank groups.

Findings

Established a generalization for rank-two groups.

Connected zeta functions with twisted Poincaré series.

Proposed a conjecture for higher rank groups.

Abstract

In the case where $G=$SL$_{2}(F)$ for a non-archimedean local field $F$ and $\Gamma$ is a discrete torsion-free cocompact subgroup of $G$, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of $G$ by the action of $\Gamma$, and an alternating product of determinants of twisted Poincar\'e series for parabolic subgroups of the affine Weyl group of $G$. We show how this can be generalised to other split simple algebraic groups of rank two over $F$, and formulate a conjecture about how this might be generalised to groups of higher rank.