Moduli of Abelian Varieties

In this paper we generalize parts of Mumford’s theory of the equations defining abelian varieties. Using the concept of a strongly symmetric line bundle, which is weaker than Mumford’s concept of totally symmetric line bundle and is introduced here for the first time, we extend Mumford’s methods of obtaining equations to arbitrary levels and to ample strongly symmetric line bundles. The first three chapters present Mumford’s theory anew in the more general setting. In the last chapter, we demonstrate that it is feasible to work with the equations using the symmetry groups of the loci and techniques of representation theory and invariant theory. To illustrate our point of view, we derive equations due to Felix Klein and to Heinrich Burkhardt for certain moduli spaces of odd level and generalize them to arbitrary dimensions. We show that the original equations of Klein say that the modular curve of level p is the intersection of a 2-uply embedded projective space and a Grassmannian. The generalized equation of Burkhardt says that the moduli space of abelian varieties with an ample strongly symmetric line bundle of level (3, . . . , 3) lies in a certain level of the singular locus of the Hessian of a quartic invariant for Sp(Fg3) which we write explicitly. Using the equations alone, we compute the degree of the modular curve of level p from its equations and show how the rational points on the modular curve are related to rational points on Fermat curves.