On Ramanujan's τ-function, Modular Forms and Hecke Operators

Abstract

This essay is a survey on modular forms developing the theory from first principals through to research problems. We will develop hyperbolic geometry as a space on which to perform complex analysis. The vector space of modular forms will be explained as complex-valued functions with periodic-like properties. Through the 1930's work of the German mathematicians Hecke and his student Petersson, we will develop the Hecke linear operator on the space of modular forms and show this operator to be multiplicative. This gives rise to modular forms that are normalised, simultaneous eigenfunctions - the so called Hecke-forms. These results will be used to explain Ramanujan’s arithmetic observations about his Ramanujan τ-function, that he could not prove. The Petersson inner product on the vector space of Modular Forms will be constructed and we show that all Hecke operators are Hermitian with respect to that product. From this we can deduce that there is a basis of simultaneous eigenforms for the vector space of modular forms. Some applications are presented, including the representation of integers as sums of squares, Fermat's last theorem and the unsolved congruent number problem.