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On equations defining moduli of abelian varieties with endomorphisms in a totally real field. (English. Russian original) Zbl 0555.14018
Trans. Mosc. Math. Soc. 1982, No. 2, 1-46 (1982); translation from Tr. Mosk. Mat. O.-va 42, 3-49 (1981).
D. Mumford has shown how to write out the equations for the moduli scheme of abelian varieties carrying symmetric theta-structures [D. Mumford, Invent. Math. 1, 287-354 (1966); 3, 75-135; 215-244 (1967; Zbl 0219.14024)]. The moduli scheme has a tower of Hecke correspondences which classify isogenies of abelian varieties. The fixed points of correspondences represent abelian varieties with non-trivial endomorphisms. The author introduces a new notion of symmetric theta- structure which is compatible with an endomorphism of abelian variety. The main theorem gives explicit equations for the subscheme of the moduli scheme which represents the abelian varieties with endomorphisms in a totally real extension of \({\mathbb{Q}}\) compatible with the theta-structure. The exact formulations are rather long but the whole result is as explicit as possible.
Reviewer: A.N.Parshin

14K10 Algebraic moduli of abelian varieties, classification
11R80 Totally real fields
14K25 Theta functions and abelian varieties
14K15 Arithmetic ground fields for abelian varieties
14K22 Complex multiplication and abelian varieties