Famous for their important role in the proof of Fermat's Last Theorem, modular forms are analytic functions in number theory whose Fourier series encode arithmetic information. Half integral weight modular forms include some important examples, such as the generating functions for integer partitions and class numbers for imaginary quadratic number fields. The most fundamental examples are the Eisenstein series. We will compute Fourier expansions for specific half integral weight Eisenstein series, and we'll see what kinds of arithmetic functions appear in the coefficients.
Elementary number theory, complex variables, and Fourier series would all be very helpful topics to have had experience with, but I definitely would not expect students to have seen all of them. Number theory would be the most valuable.