Abstract

The aim of this work is to define modular forms modulo p and to present a result about their structure adopting two perspectives. First we deal with the case of the full modular group and we see modular forms in the classical sense through their q-expansions. In
order to generalize their characterization to level N we approach another, more intrinsic, definition of modular forms which arises from the geometry of elliptic curves. Following the work of Katz they are either functions on classes of elliptic curves with additional
data, or sections of line bundles over the modular curve. The only modular form whose q-expansion is 1 is the Hasse invariant A. Multiplication by A does not change q-expansions and naturally determines a filtration on the graded algebra of modular forms. The main theorem analyzes the properties of such a filtration through the behaviour of the operator θ whose construction represents the core of this thesis.