1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 15

the Main Theorem of Complex Multiplication) are discussed in Chapter 2, and

Appendix A provides proofs of the Main Theorem of Complex Multiplication and

some results of Tate over finite fields.

1.2. Simplicity, isotypicity, and endomorphism algebras

1.2.1. Simple abelian varieties. An abelian variety A over a field K is simple

(over K) if it is non-zero and contains no non-zero proper abelian subvarieties.

Simplicity is not generally preserved under extension of the base field; see Example

1.6.3 for some two-dimensional examples over finite fields and over Q. An abelian

variety A over K is absolutely simple (over K) if AK is simple.

1.2.1.1. Lemma. If A is absolutely simple over a field K then for any field ex-

tension K /K, the abelian variety AK over K is simple.

Proof. This is an application of direct limit and specialization arguments, as we

now explain. Assume for some K /K that there is a non-zero proper abelian sub-

variety B ⊂ AK . By replacing K with an algebraic closure we may arrange that

K and then especially K is algebraically closed. (The algebraically closed property

for K is unimportant, but it is crucial that we have it for K.) By expressing K

as a direct limit of finitely generated K-subalgebras, there is a finitely generated

K-subalgebra R ⊂ K such that B = BK for an abelian scheme B → Spec(R)

that is a closed R-subgroup of AR.

The constant positive dimension of the fibers of B → Spec(R) is strictly less

than dim(A), as we may check using the K -fiber B ⊂ AK . Since K is algebraically

closed we can choose a K-point x of Spec(R). The fiber Bx is a non-zero proper

abelian subvariety of A, contrary to the simplicity of A over K.

For a pair of abelian varieties A and B over a field K,

Hom0(AK

, BK ) can be

strictly larger than

Hom0(A,

B) for some separable algebraic extension K /K. For

example, if E is an elliptic curve over Q then considerations with the tangent line

over Q force

End0(E)

= Q, but it can happen that

End0(EL)

= L for an imaginary

quadratic field L (e.g., E :

y2

=

x3

− x and L = Q(

√

−1)).

Scalar extension from number fields to C or from an imperfect field to its perfect

closure are useful techniques in the study of abelian varieties, so there is natural

interest in considering ground field extensions that are not separable algebraic (e.g.,

non-algebraic or purely inseparable). It is an important fact that allowing such

general extensions of the base field does not lead to more homomorphisms:

1.2.1.2. Lemma (Chow). Let K /K be an extension of fields that is primary (i.e.,

K is separably algebraically closed in K ). For abelian varieties A and B over K,

the natural map Hom(A, B) → Hom(AK , BK ) is bijective.

Proof. See [23, Thm. 3.19] for a proof using faithfully flat descent (which is

reviewed at the beginning of [23, §3]). An alternative proof is to show that the

locally finite type Hom-scheme Hom(A, B) over K is ´etale.

We shall be interested in certain commutative rings acting faithfully on abelian

varieties, so we need non-trivial information about the structure of endomorphism

algebras of abelian varieties. The study of such rings rests on the following funda-

mental result.