Duke for Drinfeld

We prove a function field analogue of Duke’s equidistribution theorem for CM points, in the setting of Drinfeld–Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke’s theorem on the modular curve and S.-W. Zhang’s equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger’s formula expresses through central values of automorphic $L$-functions, bounded in Lindelöf-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.