Moduli Spaces of Abelian Varieties with Endomorphism Structure

Abstract

In Chapter 5 we saw that the endomorphism algebra of a simple abelian variety is a skew field F of finite dimension over ℚ admitting a positive anti-involution′, the Rosati involution. Moreover, in Section 5.5 we classified all such pairs (F, ′). In this chapter we study the converse question: which of the pairs (F, ′) actually occur as the endomorphism algebra of a polarized abelian variety? To be more precise, for every pair (F, ′) we construct families of polarized abelian varieties (X, H) together with an endomorphism structure. Roughly speaking, this is an embedding F ↪ End _ℚ(X). The parameter spaces of these families are certain complex manifolds H, which are generalizations of Siegel’s upper half space. The families themselves are parametrized by pairs (M, T), where M is a certain ℤ-submodule of the left F-vector space F ^m and T a nondegenerate skew-hermitian form on F ^m. Moreover we analyze, when two elements of H represent isomorphic polarized abelian varieties with endomorphism structure. It turns out that, given (M, T), there is a group G(M, T) acting properly and discontinuously on H, such that the quotient A(M, T) = H/G(M, T) is a moduli space (in the sense of Chapter 8) for the polarized abelian varieties with endomorphism structure associated to (M, T). Finally we show that any polarized abelian variety with endomorphism structure is represented in one of these moduli spaces and that for a general element of A(M, T) the endomorphism algebra equals the skew field F, except for some particular cases. This answers the question we started with.