THE GEOMETRY AND ARITHMETIC OF A CALABI-YAU SIEGEL THREEFOLD

In this paper we treat in details a Siegel modular variety Y that has a Calabi-Yau model, (Y) over tilde. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of Y as the quotient of another known Calabi-Yau variety X. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution (X) over tilde of X. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi-Yau model (Y) over tilde and computing the Picard group and the Euler characteristic.