1.3. COMPLEX MULTIPLICATION 23

AK are obtained by taking A to be an elliptic curve over C with analytic model

C/(Z ⊕ Zτ) for τ ∈ C − R such that 1,τ, τ, ττ are Q-linearly independent. (In

Example 1.6.4 we give examples with K = Q.)

In cases when AK is non-isotypic, A is necessarily simple. Indeed, if A is not

simple then a simple factor of A would be a K-descent of a member of the isogeny

class of A , contradicting that A and σ∗(A ) are not isogenous. Thus, we have

exhibited examples in characteristic 0 for which isotypicity is lost after a ground

field extension.

The failure of isotypicity to be preserved after a ground field extension does

not occur over finite fields:

1.2.6.1. Proposition. If A is an isotypic abelian variety over a finite field K then

AK is isotypic for any extension field K /K.

Proof. By the Poincar´ e reducibility theorem, it is equivalent to show that

End0(AK

) is a simple Q-algebra, so by Lemma 1.2.1.2 we may replace K with

the algebraic closure of K in K . That is, we can assume that K /K is algebraic.

Writing K = lim

− →

Ki with {Ki} denoting the directed system of subfields of finite

degree over K, we have End(AK ) = lim

− →

End(AK

i

). But End(AK ) is finitely gen-

erated as a Z-module, so for large enough i we have

End0(AK

) =

End0(AK

i

). We

may therefore replace K with Ki for suﬃciently large i to reduce to the case when

K /K is of finite degree. Let q = #K.

The key point is to show that for any abelian variety B over K and any

g ∈ Gal(K /K), B and

g∗(B

) are isogenous. Since Gal(K /K) is generated by

the q-Frobenius σq, it suﬃces to show that B and B

(q)

:= σq

∗(B

) are isogenous.

The purely inseparable relative q-Frobenius morphism B → B

(q)

(arising from

the absolute q-Frobenius map B → B over the q-Frobenius of Spec(K )) is such

an isogeny. Hence, the Weil restriction ResK

/K

(B ) satisfies ResK

/K

(B )K

g

g∗(B

) ∼ B

[K :K]

.

Take B to be a simple factor of AK (up to isogeny), so ResK

/K

(B ) is an

isogeny factor of ResK

/K

(AK ) ∼

A[K :K].

By the simplicity of A and the Poincar´e

reducibility theorem, it follows that ResK

/K

(B ) is isogenous to a power of A.

Extending scalars, ResK

/K

(B )K is therefore isogenous to a power of AK . But

ResK

/K

(B )K ∼ B

[K :K]

, so non-trivial powers of AK and B are isogenous. By

the simplicity of B and Poincar´ e reducibility, this forces B to be the only simple

factor of AK (up to isogeny), so AK is isotypic.

1.3. Complex multiplication

1.3.1. Commutative subrings of endomorphism algebras. The following

fact motivates the study of complex multiplication in the sense that we shall con-

sider.

1.3.1.1. Theorem. Let A be an abelian variety over a field K with g := dim(A)

0, and let P ⊂

End0(A)

be a commutative semisimple Q-subalgebra. Then [P : Q]

2g, and if equality holds then P is its own centralizer in

End0(A).

If equality holds