The Eichler cohomology theorem

Abstract

Modular forms are associated with many problems that arise in mathematics and physics. They attain their importance due to the following properties, finite dimensionality of the space of entire modular forms Mk , rationality of Fourier coefficients of the forms that span Mk for an integer weight k, and rationality of periods. This thesis presents a thorough introduction on modular forms and their properties. We proceed to define period polynomials that emerge when defining the Eichler integrals. Eichler integral results from integrating a weight k + 2 modular form and is a function that exhibits similar properties of modular forms but has an additional factor (period polynomial) in its transformation law. These period polynomials with restricting conditions form the Eichler cohomology group. In this thesis, we study the action of Hecke operator on period polynomials. We then continue to prove the Eichler isomorphism theorem which describes the relationship between the Eichler cohomology group and spaces of entire and cusp forms.