16 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.2.1.3. Theorem (Poincar´ e reducibility). Let A be an abelian variety over a field

K. For any abelian subvariety B ⊂ A, there is is abelian subvariety B ⊂ A such

that the multiplication map B × B → A is an isogeny.

In particular, if A = 0 then there exist pairwise non-isogenous simple abelian

varieties C1,...,Cs over K such that A is isogenous to Ci

ei

for some ei 1.

Proof. When K is algebraically closed this result is proved in [82, §19, Thm. 1].

The same method works for perfect K, as explained in [76, Prop. 12.1]. (Perfectness

is implicit in the property that the underlying reduced scheme of a finite type K-

group is a K-subgroup. For a counterexample over any imperfect field, see [25,

Ex. A.3.8].) The general case can be pulled down from the perfect closure via

Lemma 1.2.1.2; see the proof of [23, Cor. 3.20] for the argument.

1.2.1.4. Corollary. For a non-zero abelian variety A over a field K and a primary

extension of fields K /K, every abelian subvariety B of AK has the form BK for

a unique abelian subvariety B ⊂ A.

Proof. By the Poincar´ e reducibility theorem, abelian subvarieties of A are pre-

cisely the images of maps A → A, and similarly for AK . Since scalar extension

commutes with the formation of images, the assertion is reduced to the bijectivity

of End(A) → End(AK ), which follows from Lemma 1.2.1.2.

Since any map between simple abelian varieties over K is either 0 or an isogeny,

by general categorical arguments the collection of Ci’s (up to isogeny) in the

Poincar´ e reducibility theorem is unique up to rearrangement, and the multiplic-

ities ei are also uniquely determined.

1.2.1.5. Definition. The Ci’s in the Poincar´ e reducibility theorem (considered

up to isogeny) are the simple factors of A.

By the uniqueness of the simple factors up to isogeny, we deduce:

1.2.1.6. Corollary. Let A be a non-zero abelian variety over a field, with simple

factors C1,...,Cs. The non-zero abelian subvarieties of A are generated by the

images of maps Ci → A from the simple factors.

1.2.2. Central simple algebras. Using notation from the Poincar´ e reducibility

theorem, for a non-zero abelian variety A we have

End0(A)

Matei

(End0(Ci))

where {Ci} is the set of simple factors of A and the ei’s are the corresponding

multiplicities. Each

End0(Ci)

is a division algebra, by simplicity of the Ci’s. Thus,

to understand the structure of endomorphism algebras of abelian varieties we need

to understand matrix algebras over division algebras, especially those of finite di-

mension over Q. We therefore next review some general facts about such rings.

Although we have used K to denote the ground field for abelian varieties above,

in what follows we will use K to denote the ground field for central simple algebras;

the two are certainly not to be confused, since for abelian varieties in positive

characteristic the endomorphism algebras are over fields of characteristic 0.