Tutor / director / evaluatorQuer Bosor, Jordi
Document typeBachelor thesis
Rights accessOpen Access
Modular forms are holomorphic functions defined on the complex upper half-plane which transform in a certain way under the action of a group of matrices. Alternatively, they can be thought of as differential forms on a Riemann surface known as a modular curve. There is a very important family of operators acting on the space of modular forms, the Hecke operators. One of their main properties is that there exist bases of spaces of modular forms consisting of eigenvectors of most Hecke operators. Finally, modular symbols can be regarded as formal symbols satisfying certain algebraic relations and which provide a simple way to represent the elements of the first homology group of modular curves. The pairing given by integration of a form along a path provides a duality between modular forms and modular symbols. Therefore, Hecke operators also act on the space of modular symbols. One can recover information about the modular forms from the action of Hecke operators on the modular symbols. In conclusion, modular symbols constitute an appropriate setting to perform computations with modular forms and Hecke operators.