A Ramanujan bound for Drinfeld modular forms

We prove a Lefschetz trace formula for Böckle-Pink crystals on tame Deligne-Mumford stacks of finite type over ${\mathbb{F}}_q$ and apply it to the crystal associated to the universal Drinfeld module. Combined with the Eichler-Shimura theory developed by Böckle, this leads to a trace formula for Hecke operators on Drinfeld modular forms. As an application, we deduce a Ramanujan bound on the traces of Hecke operators.