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to reconstruct this (unpublished) proof by Mumford.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

1Of

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.

2The

prime number is assumed to be invertible in the base field.

1

http://dx.doi.org/10.1090/surv/195/01 http://dx.doi.org/10.1090/surv/195/01