Impacted Buildings for GL(2)

TL;DR

This paper introduces a generating function for certain filtered trees related to GL(2) buildings, linking it to the zeta functions of quadratic orders through ideal distribution analysis.

Contribution

It defines a new generating function for filtered $ ilde{A}_1$ buildings and connects it to quadratic order zeta functions, providing a novel analytical approach.

Findings

The generating function recovers the zeta function of quadratic orders.

Ideal distribution in $SL(2, K)$ buildings is characterized.

New connections between building theory and number theory are established.

Abstract

In this paper we define a generating function for buildings of type $\widetilde{A}_1$ (i.e. trees) that are enhanced with a certain filtration structure. We prove that this generating function recovers the zeta function of certain quadratic orders. We do this by studying how the ideals of the orders distribute in the building of $SL(2, K)$.