| Summary: | The purpose of this thesis is to study the number theoretic properties of real analytic modular forms, which were recently introduced by F. Brown. We begin with a brief review of the classical theory of modular forms, focusing on mathematical objects such as Hecke operators, L-functions and period polynomials.
In Chapter 2, we introduce and discuss real analytic modular forms. We define L-functions for the entirety of this space and establish their main properties. We then define the analogue of the period polynomial for modular iterated integrals of length one.
In Chapter 3, we review the action of Hecke operators on the period polynomials of standard modular forms, with the aim of extending this theory to real analytic modular forms. We achieve this for a certain type of length one modular iterated integral called a real analytic cusp form.
In the final chapter, we present a theorem for producing new and interesting length three modular iterated integrals. This can be viewed as an extension of the length two case given by Brown, a review of which is included in this chapter. We discuss how modular iterated integrals could help understand modular graph functions, which arise in string perturbation theory.
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