The surface of cuboids and Siegel modular threefolds

A perfect cuboid is a parallelepiped with rectangular faces all of whose edges, face diagonals and long diagonal have integer length.  A question going back to Euler asks for the existence of a perfect cuboid. 
No perfect cuboid has been found, nor it is known that they do not exist.

In this talk I will talk about Siegel modular threefolds: these are certain moduli spaces of abelian surfaces with level structure.  Then, I will proceed to show that the space of cuboids is a divisor in a one of these moduli spaces.  Therefore the existence of a perfect cuboid is equivalent to the existence of special torsion structures in abelian surfaces defined over number fields.