Quotients of the Bruhat-Tits tree by function field analogs of the Hecke congruence 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 this work we describe the associated quotient graph H﹨t for the action on t of the group H={(abcd)∈GL2(R):c≡0(mod I)}, where I is an ideal of R. These groups play, in the function field context, the same role as the Hecke congruence subgroups of SL2(Z). To be precise, we give a explicit formula for the cusp number of H﹨t. Then, by using Bass-Serre Theory, we describe H as an amalgamated product of simpler groups, and we also describe its abelianization.