A Ramanujan bound for Drinfeld modular forms

TL;DR

This paper establishes a trace formula for Hecke operators on Drinfeld modular forms and derives a Ramanujan bound, extending classical bounds to the function field setting.

Contribution

It introduces a Lefschetz trace formula for B"ockle-Pink crystals and applies it to Drinfeld modules to obtain new bounds on Hecke operator traces.

Findings

Proved a Lefschetz trace formula for B"ockle-Pink crystals.

Derived a Ramanujan bound for traces of Hecke operators.

Extended classical bounds to the context of Drinfeld modular forms.

Abstract

We prove a Lefschetz trace formula for B\"ockle-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 B\"ockle, 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.