Rational Points on Drinfeld Modular Curves of Prime Level

Let $q\ge 3$ be a prime power and let $\mathfrak{p}\subset {\mathbf{F}}_q[T]$ be a nonzero prime ideal of degree $d$. We prove that $Y_0(\mathfrak{p})({\mathbf{F}}_q(T))=\mathrm{\varnothing }$ if $d\ge 4$ is even or $d>q$, and that $Y_1(\mathfrak{p})({\mathbf{F}}_q(T))=\mathrm{\varnothing }$ for every $d\ge 3$.