On algebraic families of polarized abelian varieties (Q1969659)

Let \(L\subset M_{2g} (\mathbb{Q})\) be an admissible Albert division algebra, that is an algebra which is the endomorphism algebra of some principally polarized abelian variety of dimension \(g\). \(L\) defines a subvariety \(\Gamma(L) \setminus \mathbb{H}(L)\) of \(Sp(2g,\mathbb{Z})\setminus\mathbb{H}_g\), the moduli space of principally polarized abelian varieties, parametrizing abelian varieties whose endomorphism algebra contains \(L\). Denote by \(C(L)\) its closure in the Satake compactification of \(Sp(2g, \mathbb{Z})\setminus \mathbb{H}_g\), and by \({\mathcal A}(L)\) its normalization. The varieties \(\Gamma(L) \setminus \mathbb{H}(L)\) and \({\mathcal A}(L)\) are called Shimura varieties. In this note such Shimura varieties are studied using a real model of the Siegel upper half space.