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
(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to reconstruct this (unpublished) proof by Mumford.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  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 :
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  and ), and Tate refined this by describing
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.
prime number is assumed to be invertible in the base field.