Parameter spaces for complex tori with endomorphism structure (Q1383581)

In a famous paper Shimura constructed families of polarized abelian varieties with endomorphism structure. These families are parametrized by products of bounded symmetric domains of types C I (the Siegel upper half space), A III and D III. In this note the authors generalize some of Shimura's constructions to nondegenerate complex tori. A nondegenerate complex torus is a pair consisting of a complex torus \(X\) and a class \(H\) of nondegenerate line bundles on \(X\). The authors give necessary conditions for a skewfield to be the endomorphism algebra of a nondegenerate complex torus. They present the spaces parametrizing families of nondegenerate complex tori with endomorphism structure in these skewfields. The parameter spaces are flag domains associated to classical groups in the sense of J. Wolf.