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174 Chapter 6. Complex multiplication in higher generaEquivalently, E r is generated by the elements of the form∑φ(a), a ∈ E .φ∈ΦThe reflex field is a CM field and is left invariant by extension of CM algebra and type;moreover, if (E, Φ) = ∏ ni=1 (E i, Φ i ), then E r = E1 r · · · En r [205, Proposition I.1.18].If F is a CM field and Φ a CM type, then Φ can be extended to the Galois closure L of F .Denote by (K, Ψ) the primitive CM subpair of (L, Φ −1 ). Then K = E r and Ψ is called the reflexCM type of Φ [204, Example I.1.19], [159, Theorem 1.5.1].Definition 6.3.6 (Type trace [159, 1.5]). Let (E, Φ) be a CM pair and E r its reflex field. Thetype trace Tr Φ is defined asE → E r ,x ↦→ ∑ φ∈Φφ(x) .Definition 6.3.7 (Type norm [159, 1.5], [205, I.1.5]). Let (E, Φ) be a CM pair and E r its reflexfield. The type norm N Φ is defined asE → E r ,x ↦→ ∏ φ∈Φφ(x) .The type norm naturally extends to fractional ideals and induces a map between class groups.6.3.3 CM abelian varietiesDefinition 6.3.8 (CM abelian variety [205, I.3]). Let A be an abelian variety of dimension g. Wesay that A is an abelian variety with complex multiplication or is a CM abelian variety 11 if thereexists a commutative semisimple Q-algebra E of dimension 2g and an embedding i : E → End 0 (A).The CM abelian variety A, or the pair (A, i), is also said to have complex multiplication byE or R = i −1 (End 0 (A)) an order in E. If (A, i) has complex multiplication by R, then we saythat it is defined over k, or that (A, i) has complex multiplication over k, if A is and if everyhomomorphism i(R) is. If R is the maximal order of E, then we say that A is principal.Using the decomposition of A into a product of simple abelian varieties it can be shown thatan abelian variety has complex multiplication if and only if each simple factor has. Moreover,the endomorphism algebra of a simple abelian variety is a division algebra, so it has complexmultiplication if and only if it contains a field of degree 2g.Tate has shown that every abelian variety defined over a finite field has complex multiplication[261].Not only over the complex numbers, but over any field, the maximal commutative algebracan be chosen to be a CM algebra [286].Over the complex numbers, a CM abelian variety is then characterized as follows.Proposition 6.3.9 ([205, Proposition I.3.6]). Let A be a complex abelian variety of dimension g:• If A is simple, then it has complex multiplication if and only if its endomorphism algebra isa CM field K.11 One also says that the abelian variety has sufficiently many complex multiplication or is of CM type.