Giraud, Jean Modules des variétés abéliennes et variétés de Shimura. (French) Zbl 0538.14030 Variétés de Shimura et fonctions L, Publ. Math. Univ. Paris VII, No. 6, 21-42 (1979). [For the entire collection see Zbl 0517.00003.] This seminar talk claims no originality but is a clean and explicit exposition. It starts with a description of the objects studied in the seminar namely abelian varieties X provided with an action of an order of a quaternion algebra B over a totally real number field E such that \(\dim X=g=2(E:{\mathbb{Q}})\). A review of the structure and properties of End(X) is first given. Then a universal family is carefully constructed from which a coarse moduli space is derived. All of this in the complex analytic framework. In § 3, the adelic interpretation is given. Cited in 4 Documents MSC: 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14D22 Fine and coarse moduli spaces 14K10 Algebraic moduli of abelian varieties, classification 14L30 Group actions on varieties or schemes (quotients) 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) 12E15 Skew fields, division rings Keywords:Shimura variety; Abelian varieties with action of order of quaternion algebra; coarse moduli space Citations:Zbl 0517.00003 PDFBibTeX XML