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In w 1 we consider the situation: L/K is a finite separable field extension, A is an abelian variety over L, and A, is the abelian variety over K obtained from A by restriction of scalars. We study the arithmetic properties of A, relative to those of A, and in particular show that the conjectures of Birch and Swinnerton-Dyer hold for A if and only if they hold for A,.In w 2 we study certain twisted products of abelian varieties and use our results to show that the conjectures of Birch and Swinnerton-Dyer are true for a large class of twisted constant elliptic curves over function fields.In w we develop a method of handling abelian varieties over a number field K which are of CM-type but which do not have all their complex multiplications defined over K. In particular we compute under quite general conditions the conductors and zeta functions of such abelian varieties and so verify Serre's conjecture [12] w 1. The Arithmetic lnvariants of the NormLet T--, S be a morphism of schemes. We recall the definition and properties of the norm functor NT/s (in [19] this is denoted by RT/s and called restriction of field of definition, and in [3, Exp. 195] it is denoted by FIT/s). If X is a T-scheme then NT/sX is uniquely determined as the S-scheme which represents the functor on S-schemes Z ~ X(ZT), where Z T =Z • T. There is a T-morphism p: (NT/sX) T --~ X such that any other T-morphism p': ZT~ X factors uniquely as P'=PqT with q: Z---, NT/sX an S-morphism. NT./sX always exists if X is quasi-projective and T-~ S is finite and faithfully flat [3, Exp. 221], and it is obvious from the definition that NT/s commutes with base change on S. If X is a group scheme then NT/sX acquires a unique group structure such that p is a morphism of group schemes. If X is smooth over T then it is obvious from the

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