26 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Mate(D) is the product of the Z-degrees of L and the centralizer of L in Mate(D).
But L is its own centralizer, so
e2[D
: Z] = dimZ Mate(D) = [L :
Z]2
=
e2(2d/[Z
:
Q])2.
We conclude that 2d/[Z : Q] = [D : Z], so (by Proposition 1.2.3.1) 2d/[Z : Q] is
the common Z-degree of all maximal commutative subfields of the central division
algebra D =
End0(C)
over Z, or equivalently 2d is the Q-degree of all such fields.
But 2d = 2 dim(C), so choosing any maximal commutative subfield of D shows
that C admits sufficiently many complex multiplications.
1.3.3. CM algebras and CM abelian varieties. The following three conditions
on a number field L are equivalent:
(1) L has no real embeddings but is quadratic over a totally real subfield,
(2) for every embedding j : L C, the subfield j(L) C is stable under complex
conjugation and the involution x
j−1(j(x))
in Aut(L) is non-trivial and
independent of j,
(3) there is a non-trivial involution τ Aut(L) such that for every embedding
j : L C we have j(τ(x)) = j(x) for all x L.
The proof of the equivalence is easy. When these conditions hold, τ in (3) is unique
and its fixed field is the maximal totally real subfield
L+
L (over which L is
quadratic). The case
L+
= Q corresponds to the case when L is an imaginary
quadratic field.
1.3.3.1. Definition. A CM field is a number field L satisfying the equivalent
conditions (1), (2), and (3) above. A CM algebra is a product L1 × · · · × Ls of
finitely many CM fields (with s 1).
The reason for this terminology is due to the following important result (along
with Example 1.5.3).
1.3.4. Theorem (Tate). Let A be an abelian variety of dimension g 0 over a
field K. Suppose A admits sufficiently many complex multiplications. Then there
exists a CM algebra P
End0(A)
with [P : Q] = 2 dim(A). In case A is isotypic
we can take P to be a CM field.
The proof of this theorem (which ends with the proof of Lemma 1.3.7.1) will
require some effort, especially since we consider an arbitrary base field K. Before
we start the proof, it is instructive to consider an example.
1.3.4.1. Example. Consider A =
E2
with an elliptic curve E over K = C such
that L :=
End0(E)
is an imaginary quadratic field. The endomorphism algebra
End0(A)
= Mat2(L) is simple and contains as its maximal commutative subfields
all quadratic extensions of L. Those extensions which are biquadratic over Q are
CM fields, and the rest are not CM fields. Hence, in the setup of Theorem 1.3.4, even
when A is isotypic and char(K) = 0 there can be maximal commutative semisimple
subalgebras of
End0(A)
that are not CM algebras. However, if char(K) = 0 and A
is simple (over K) then
End0(A)
is a CM field; see Proposition 1.3.6.4.
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