Modular Forms, Galois Representations, and Cohomology of Line Bundles

Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a vector bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.