Explicit Arithmetic of Hilbert Modular Surfaces

希尔伯特模曲面是数论中的一个经典领域可以追溯到20世纪初。这些模空间对具有特定自同态环和水平结构的超几何阿贝尔曲面进行分类。在一篇重要的论文中，Elkies和Kummar给出了不含水平结构的Hilbert模曲面的显式方程。考虑到模块化转换的应用（例如在Ellenberg的工作中），具有水平结构的希尔伯特模块化曲面的显式方程是很重要的，这是该项目的目的。

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.