Classical modular forms and modular curves are fundamental to contemporary research in number theory, as they bring together many different areas of study, including modular forms, elliptic curves, arithmetic q-series, and class field theory. The PI's propose to investigate new problems in the theory of Drinfeld modular forms in the setting of global function fields in finite characteristic. Drinfeld modular forms are rigid analytic functions on the upper half plane that transform naturally with respect to the action of the modular group over the polynomial ring over a finite field, and they induce differential forms on the corresponding Drinfeld modular curves. The PI's plan to consider a number of problems related to Drinfeld modular forms and Drinfeld modular curves, with an eye toward improved understanding of the classical theory. They will investigate equidistribution properties of Heegner points, which arise from the theory of Drinfeld modules with complex multiplication, as well as associated problems for higher degree Drinfeld modular forms of Hilbert type. They will investigate new problems about the arithmetic properties of traces of singular moduli on Drinfeld modular curves, and the connection between integral points associated to non-split Cartan subgroups and the class number one problem. Furthermore, they will research problems on Fourier expansions of vector valued Drinfeld modular forms and associated L-serie
