Arithmetic properties of modular forms and the Weil representation

The Fourier coefficients of modular forms are rich in arithmetic content. For example, we know that their coefficients are related to special values of L-series, representation numbers of quadratic forms, and generalized divisor functions. In this thesis, we discuss some results on the arithmeticity of various types of modular forms.;In a recent paper [2], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. In Part II of this thesis, we give an affirmative answer to Borcherds' question. This is an analogue of a classical result, due to Shimura, for scalar valued modular forms associated to congruence subgroups.;In Part III, we discuss a result of Ono and Skinner [15] which suggests that there may be a "mod p" theory of half integral weight modular forms. Specifically, the authors are able to show that, if a half-integral weight eigenform g is good, then g has infinitely many coefficients prime to ℓ, for all but finitely many primes ℓ. Their paper ends with the conjecture that all half-integral weight modular forms (with the exception of certain theta functions) are good. We give a proof of this conjecture.