AK are obtained by taking A to be an elliptic curve over C with analytic model
C/(Z Zτ) for τ C R such that 1,τ, τ, ττ are Q-linearly independent. (In
Example 1.6.4 we give examples with K = Q.)
In cases when AK is non-isotypic, A is necessarily simple. Indeed, if A is not
simple then a simple factor of A would be a K-descent of a member of the isogeny
class of A , contradicting that A and σ∗(A ) are not isogenous. Thus, we have
exhibited examples in characteristic 0 for which isotypicity is lost after a ground
field extension.
The failure of isotypicity to be preserved after a ground field extension does
not occur over finite fields: Proposition. If A is an isotypic abelian variety over a finite field K then
AK is isotypic for any extension field K /K.
Proof. By the Poincar´ e reducibility theorem, it is equivalent to show that
) is a simple Q-algebra, so by Lemma we may replace K with
the algebraic closure of K in K . That is, we can assume that K /K is algebraic.
Writing K = lim

Ki with {Ki} denoting the directed system of subfields of finite
degree over K, we have End(AK ) = lim

). But End(AK ) is finitely gen-
erated as a Z-module, so for large enough i we have
) =
). We
may therefore replace K with Ki for sufficiently large i to reduce to the case when
K /K is of finite degree. Let q = #K.
The key point is to show that for any abelian variety B over K and any
g Gal(K /K), B and
) are isogenous. Since Gal(K /K) is generated by
the q-Frobenius σq, it suffices to show that B and B
:= σq
) are isogenous.
The purely inseparable relative q-Frobenius morphism B B
(arising from
the absolute q-Frobenius map B B over the q-Frobenius of Spec(K )) is such
an isogeny. Hence, the Weil restriction ResK
(B ) satisfies ResK
(B )K
) B
[K :K]
Take B to be a simple factor of AK (up to isogeny), so ResK
(B ) is an
isogeny factor of ResK
(AK )
A[K :K].
By the simplicity of A and the Poincar´e
reducibility theorem, it follows that ResK
(B ) is isogenous to a power of A.
Extending scalars, ResK
(B )K is therefore isogenous to a power of AK . But
(B )K B
[K :K]
, so non-trivial powers of AK and B are isogenous. By
the simplicity of B and Poincar´ e reducibility, this forces B to be the only simple
factor of AK (up to isogeny), so AK is isotypic.
1.3. Complex multiplication
1.3.1. Commutative subrings of endomorphism algebras. The following
fact motivates the study of complex multiplication in the sense that we shall con-
sider. Theorem. Let A be an abelian variety over a field K with g := dim(A)
0, and let P
be a commutative semisimple Q-subalgebra. Then [P : Q]
2g, and if equality holds then P is its own centralizer in
If equality holds
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