Realizing Hecke Actions on Modular Forms Via Cohomology of Dessins d’Enfants

Abstract

A well known action on the space of Modular forms is done by Hecke operators, which also play an important role in modularity. This action can be further break down into steps, and form what we call a Hecke correspondence. On the other hand, through Belyi and Grothendieck there is a one to one correspondence between the equivalence classes of algebraic curves defined over $\Q$ equipped with Belyi functions and equivalence classes of dessins d'enfants. This applies in particular to modular curves. In my dissertation work, I will study the action of Hecke operators on certain dessins, namely, those that correspond to $X_0(N)$. This is done by defining a cohomology with coefficients in a local system on dessins, and have Hecke operators act on it. We will also construct a Hecke-equivalent isomorphism of the cohomology group with the space of cusp forms. We hope that this work can present the first step in studying the Hecke action on more general dessins.