16 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 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
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; see the proof of [23, Cor. 3.20] for the argument. 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
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. 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: 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
where {Ci} is the set of simple factors of A and the ei’s are the corresponding
multiplicities. Each
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.
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