Introduction I restricted myself to characteristic zero: for a short time, the quantum jump to p = 0 was beyond the range . . . but it did not take me too long to make this jump. — Oscar Zariski The arithmetic of abelian varieties with complex multiplication over a number field is fascinating. However this will not be our focus. We study the theory of complex multiplication in mixed characteristic. Abelian varieties over finite fields. In 1940 Deuring showed that an elliptic curve over a finite field can have an endomorphism algebra of rank 4 [33, §2.10]. For an elliptic curve in characteristic zero with an endomorphism algebra of rank 2 (rather than rank 1, as in the “generic” case), the j-invariant is called a singular j- invariant. For this reason elliptic curves with even more endomorphisms, in positive characteristic, are called supersingular.1 Mumford observed as a consequence of results of Deuring that for any elliptic curves E1 and E2 over a finite field κ of characteristic p 0 and any prime = p, the natural map Z ⊗Z Hom(E1,E2)−→ HomZ [Gal(κ/κ)] (T (E1),T (E2)) (where on the left side we consider only homomorphisms “defined over κ”) is an isomorphism [118, §1]. The interested reader might find it an instructive exercise to reconstruct this (unpublished) proof by Mumford. Tate proved in [118] that the analogous result holds for all abelian varieties over a finite field and he also incorporated the case = p by using p-divisible groups. He generalized this result into his influential conjecture [117]: An -adic cohomology class2 that is fixed under the Galois group should be a Q -linear combination of fundamental classes of algebraic cycles when the ground field is finitely generated over its prime field. Honda and Tate gave a classification of isogeny classes of simple abelian vari- eties A over a finite field κ (see [50] and [121]), and Tate refined this by describing 1 Of course, a supersingular elliptic curve isn’t singular. A purist perhaps would like to say “an elliptic curve with supersingular j-value”. However we will adopt the generally used terminology “supersingular elliptic curve” instead. 2 The prime number is assumed to be invertible in the base field. 1 http://dx.doi.org/10.1090/surv/195/01

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