Duke for Drinfeld

Abstract

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.