Involutory elliptic curves over \(\mathbb{F}_q(T)\) (Q1273182)

Let \(\mathbb{F}_q\) be a field with \(q\) elements, \(n\in\mathbb{F}_q[T]-\{0\}\) and \(G\) be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve \(X_0(n)\). The author determines all \(n\) and \(G\) such that the quotient curve \(G\setminus X_0(n)\) is rational or elliptic. This problem is related, as suggested in the title, to the classical (and solved) problem on involutory elliptic curves. Interesting calculations about the genus of \(G\setminus X_0(n)\) are done in \S 2.