Let $$K$$ K be an algebraic function field with constant field $${\mathbb {F}}_q$$ F q . Fix a place $$\infty $$ ∞ of $$K$$ K of degree $$\delta $$ δ and let $$A$$ A be the ring of elements of $$K$$ K that are integral outside $$\infty $$ ∞ . We give an explicit description of the elliptic points for the action of the Drinfeld modular group $$G=GL_2(A)$$ G = G L 2 ( A ) on the Drinfeld’s upper half-plane $$\Omega $$ Ω and on the Drinfeld modular curve $$G{\setminus }\Omega $$ G \ Ω . It is known that under the building map elliptic points are mapped onto vertices of the Bruhat–Tits tree of $$G$$ G . We show how such vertices can be determined by a simple condition on their stabilizers. Finally for the special case $$\delta =1$$ δ = 1 we obtain from this a surprising free product decomposition for $$PGL_2(A)$$ P G L 2 ( A ) .