UC Berkeley

The purpose of this thesis is to outline the Hecke stability method (HSM), a novel method for the computation of modular forms.

The HSM relies on the following idea: A finite-dimensional space of ratios of modular forms that is stable under the action of a Hecke operator should consist of modular forms (i.e., without poles). This principle is correct over the complex numbers, but more care is required over finite fields due to complications arising near the supersingular points on modular curves. Formalizing this main idea as a theorem comprises most of our theoretical work.

Though it can be utilized in a variety of settings, the main application of the Hecke stability method is the computation of weight 1 modular forms. These spaces cannot be computed using the algorithms (e.g., modular symbols algorithms) that are typically employed to compute modular forms of higher weight.

Furthermore, to provide a complete picture of the weight 1 modular forms of level N, we must account for certain sporadic discrepancies between the space of classical forms and the space of mod p modular forms. Ultimately, our approach is motivated by the effect this "ethereality" phenomenon may have on the statistics of number fields via the theory of modular Galois representations.