Rational Points on Drinfeld Modular Curves of Prime Level

Source abstract

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 $Y_0(\mathfrak{p})(\mathbf{F}_q(T))=\emptyset$ if $d \ge 4$ is even or $d>q$, and that $Y_1(\mathfrak{p})(\mathbf{F}_q(T)) = \emptyset$ for every $d \ge 3$.