Quotients of the Bruhat-Tits tree by function field analogs of the Hecke congruence subgroups

TL;DR

This paper studies the quotient graphs of certain matrix groups over function fields, extending Serre's work on Bruhat-Tits trees, and provides explicit formulas for cusp counts and the structure of these groups.

Contribution

It explicitly describes the quotient graph for groups analogous to Hecke congruence subgroups over function fields, including cusp counts and combinatorial structure.

Findings

Explicit formula for the cusp number of H\t.

Description of the combinatorial structure of H using Bass-Serre Theory.

Extension of Serre's results to function field analogs of Hecke subgroups.

Abstract

Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P of C, let R be the ring of functions that are regular outside P, and let K be the completion at P of the function field of C. In order to study groups of the form GL2(R), Serre describes the quotient graph GL2(R)\t, where t is the Bruhat-Tits tree defined from SL2(K). In particular, Serre shows that GL2(R)\t is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. It is not hard to see that finite index subgroups inherit this property. In this work we describe the associated quotient graph H\t for the action on t of the group H of matrices in GL2(R) that are upper triangular modulo a certain ideal I of R. More specifically, we give a explicit formula for the cusp number of H\t. Then, by using Bass-Serre Theory, we describe the combinatorial structure of H. These groups play, in the function field context, the same role as the Hecke congruence subgroups of SL2(Z).