HOMOMORPHISMS OF ABELIAN VARIETIES by Yuri

— We study Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications of these properties to hyperelliptic jacobians are discussed. Résumé (Homomorphismes des variétés abéliennes). — Nous étudions les propriétés galoisiennes des points d’ordre fini des variétés abéliennes qui impliquent la simplicité de leur algèbre d’endomorphismes. Nous discutons ceux-ci par rapport aux jacobiennes hyperelliptiques. It is well-known that an abelian variety is (absolutely) simple or is isogenous to a self-product of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prime order ` is “big enough”. The paper is organized as follows. In §1 we discuss Galois properties of points of order ` on an abelian variety X that imply that its endomorphism algebra End(X) is a central simple algebra over the field of rational numbers. In §2 we prove that similar Galois properties for two abelian varieties X and Y combined with the linear disjointness of the corresponding fields of definitions of points of order ` imply that X and Y are non-isogenous (and even Hom(X,Y ) = 0). In §3 we give applications to endomorphism algebras of hyperelliptic jacobians. In §4 we prove that if X admits multiplications by a number field E and the dimension of the centralizer of E in End(X) is “as large as possible” then X is an abelian variety of CM-type isogenous to a self-product of an absolutely simple abelian variety. Throughout the paper we will freely use the following observation [21, p. 174]: if an abelian variety X is isogenous to a self-product Z of an abelian variety Z then a choice of an isogeny between X and Z defines an isomorphism between End(X) and the algebra Md(End (Z)) of d × d matrices over End(Z). Since the center of 2000 Mathematics Subject Classification. — 14H40, 14K05.