Arithmetic aspects of the Burkhardt quartic threefold

The Burkhardt quartic is a 3-dimensional projective hypersurface defined over the rational numbers. It is known that sufficiently general points on the Burkhardt quartic parameterize abelian surfaces with a full level 3 structure. Furthermore, it is classical that the Burkhardt quartic is birational to 3-dimensional projective space after adjoining a cube root of unity. In this thesis we will show that the Burkhardt quartic is birational to 3-dimensional projective space over the rational numbers, and describe a geometric method of constructing a generic family of hyperelliptic curves corresponding to points on the Burkhardt quartic, whose Jacobians have a full level 3 structure. Specifically, we give an explicit family of hyperelliptic curves which contain almost all complex genus 2 curves with a full level 3 structure.