Sjoerd de Vries: The Hecke trace formula for Drinfeld modular forms

Abstract: This thesis consists of five papers on Drinfeld modular forms and their Hecke operators.

Paper I lays the foundations by investigating to what extent the traces of an operator determine its eigenvalues in positive characteristic. The ideas developed there are used, both implicitly and explicitly, throughout the remainder of the thesis.

Paper II establishes the Hecke trace formula and deduces a Ramanujan bound for Drinfeld modular forms. To this end, machinery is developed to advance the theory of crystals over function fields, culminating in a version of Behrend's trace formula for crystals on tame Deligne--Mumford stacks. Applying this to the crystal of cusp forms on the moduli space of Drinfeld modules yields the Hecke trace formula.

In Paper III, the Hecke trace formula from Paper II is applied in the special case $A = \mathbb{F}_q[T]$, where it is made as concrete and computable as possible. This leads to numerous new results, including explicit formulas for Hecke eigenvalues, computations of isogeny classes of Drinfeld modules in characteristic 2, and proofs of conjectures and open problems in the field. The resulting computational data also motivate several new conjectures.

Paper IV investigates spaces of Drinfeld quasi-modular forms. This broader setting allows for taking derivatives and hyperderivatives of Drinfeld modular forms. Several structure theorems are proved. An important conceptual advancement is the introduction of the double-slash operator, which provides a natural definition of Hecke operators on Drinfeld quasi-modular forms.

Paper V concerns traces of Hecke operators for Drinfeld modular forms as well as elliptic modular forms, modulo prime powers. The main results show that these traces are periodic in the weight, with an explicit period that works for any level. In the elliptic setting, this extends previous work of Koike, Serre, and others. The proof consists of a careful arithmetic analysis of the Hecke trace formula.