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

Abstract: Let C be a smooth, projective, and geometrically connected curve defined over a finite field F. For each closed point P_infty of C, let R be the ring of functions that are regular outside P_infty, and let K be the completion path P_infty of the function field of C. In order to study group of the form GL_2(R), Serre describes the quotient graph GL_2(R)\T, where T is the Bruhat-Tits tree defined from SL_2(K). In particular, Serre shows that GL_2(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 exposition we describe the quotient graph H\T defined from the action on T of the group H consisting of matrices that are upper triangular modulo I, where I is an ideal R. More specifically, we give an explicit formula for the cusp number 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 SL_2(Z). Moreover, not that the groups studied by Serre correspond to the case where the ideal I coincides with the ring R.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle