A Ramanujan Bound for Drinfeld Modular Forms

Abstract 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.