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
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