THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT-TITS TREE: AN ELEMENTARY APPROACH

Abstract

Let K be an algebraic function field of one variable with constant field k and let C be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place infinity of K. When k is finite the group GL(2)(C) plays a central role in the theory of Drinfeld modular curves analogous to that played by SL2(Z) in the classical theory of modular forms. When k is finite (respectively, infinite) we refer to a group GL(2)(C) as an arithmetic (respectively, non-arithmetic) Drinfeld modular group. Associated with GL(2)(C) is its Bruhat-Tits tree, T. The structure of the group is derived from that of the quotient graph GL(2)(C)\T. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of T. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of GL(2)(C)\T for the important special case where infinity has degree 1.