Explicit Arithmetic of Hilbert Modular Surfaces

Hilbert modular surfaces is a classical field within number theory going back to the early 1900s. These are moduli spaces that classify hypergeometric abelian surfaces with specified endomorphism ring and level structure. In a monumental paper, Elkies and Kummar give explicit equations for Hilbert modular surfaces without level structure and with endomorphism ring of discriminant at most 100. In view of applications to modularity switching (such as in the work of Ellenberg), it is important to have explicit equations for Hilbert modular surfaces with level structure, and this is the aim of the project.