22 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1.2.5.2. Definition. An abelian variety A over a field K is isotypic if it is isogenous

to

Ce

for a simple abelian variety C over K with e 1; that is, up to isogeny, A

has a unique simple factor. For a simple factor C of an abelian variety A over K,

the C-isotypic part of A is the isotypic subvariety of A generated by the images of

all maps C → A. An isotypic part of A is a C-isotypic part for some such C.

Clearly

End0(A)

is a semisimple Q-algebra. It is simple if and only if A is

isotypic, and it is a division algebra if and only if A is simple.

By the Poincar´ e reducibility theorem, every non-zero abelian variety A over a

field K is naturally isogenous to the product of its distinct isotypic parts, and these

distinct parts admit no non-zero maps between them. Hence, if {Bi} is the set

of isotypic parts of A then

End0(A)

=

End0(Bi)

with each

End0(Bi)

a simple

algebra of finite dimension over Q. Explicitly, if Ci is the unique simple factor of

Bi then a choice of isogeny Bi → Ci

ei

defines an isomorphism from

End0(Bi)

onto

the matrix algebra Matei

(End0(Ci))

over the division algebra

End0(Ci).

Beware

that the composite “diagonal” ring map

End0(Ci)

→ Matei

(End0(Ci)) End0(Bi)

is canonical only when

End0(Ci)

is commutative.

In general isotypicity is not preserved by extension of the ground field. To

make examples illustrating this possibility, as well as other examples in the theory

of abelian varieties, we need the operation of Weil restriction of scalars. For a

field K and finite K-algebra K , the Weil restriction functor ResK

/K

from quasi-

projective K -schemes to separated (even quasi-projective) K-schemes of finite type

is characterized by the functorial identity ResK

/K

(X )(A) = X (K ⊗K A) for

K-algebras A. Informally, Weil restriction is an algebraic analogue of viewing

a complex manifold as a real manifold with twice the dimension. In particular, if

K /K is an extension of fields then ResK

/K

(X ) is K -smooth and equidimensional

when X is K-smooth and equidimensional, with

dim(ResK

/K

(X )) = [K : K] · dim(X ).

We refer the reader to [10, §7.6] for a self-contained development of the con-

struction and properties of Weil restriction (replacing K with more general rings),

and to [25, A.5] for a discussion of further properties (especially of interest for

group schemes). In general the formation of Weil restriction naturally commutes

with any extension of the base field, and for K equal to the product ring

Kn

we

have that ResK

/K

carries a disjoint union

n

i=1

Si of quasi-projective K-schemes

(viewed as a K -scheme) to the product Si. Thus, the natural isomorphism

ResK

/K

(X )Ks Res(K

⊗K Ks)/Ks

(XK

⊗K Ks

)

implies that if K is a field separable over K then ResK

/K

(A ) is an abelian variety

over K of dimension [K : K]dim(A ) for any abelian variety A over K (since

K ⊗K Ks

KsK [ :K]

). If K /K is a field extension of finite degree that is not

separable then ResK

/K

(X ) is never proper when X is smooth and proper of

positive dimension [25, Ex. A.5.6].

1.2.6. Example. Consider a separable quadratic extension of fields K /K and a

simple abelian variety A over K . Let σ ∈ Gal(K /K) be the non-trivial element,

so K ⊗K K K × K via x ⊗ y → (xy, σ(x)y). Thus, the Weil restriction

A := ResK /K(A ) satisfies AK A ×

σ∗(A

), so AK is not isotypic if and only

if A is not isogenous to its σ-twist. Hence, for K = R examples of non-isotypic