Drinfeld singular moduli, hyperbolas, units

Let q2 be a prime power and consider Drinfeld modules of rank 2 over FqT. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas XY=, where is a polynomial of small degree. This is an effective Andr\'e-Oort theorem for these curves. We also prove that there are at most finitely many Drinfeld singular moduli that are algebraic units, for every fixed q2, and we give an effective bound on the discriminant of such singular moduli. We give in an appendix an inseparability criterion for values of some classical modular forms, generalising an argument used in the proof of our first result.