72 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
(3) The field Z is a CM field. In such cases, D is the central division algebra over
Z that is split at all places of Z away from p and for each p-adic place v of Z
has local invariant invv(D) = (ordv(π)/ordv(q))[Zv : Qp] mod Z ∈ Q/Z.
The members of the corresponding isogeny class of simple abelian varieties
over κ have dimension g = (1/2)[Z : Q] · [D : Z].
Proof. Since π is a Weil q-integer, for any embedding j : Z → C the subfield
j(Z) in C is stable under complex conjugation and the effect of this involution on Z
is given by the intrinsic formula π → q/π that is independent of j. Thus, Z is either
totally real or a CM field and the totally real cases are precisely when
= q =
with Z = Q for even n and Z = Q(
p) for odd n.
In all cases, D is split at the finite places of Z away from p. Indeed, Tate’s
isogeny theorem away from p implies that for a prime = p the map
(22.214.171.124) Q ⊗Q D → EndZ (V (A))
is an isomorphism (using that the Z-action on V (A) encodes the action of the Galois
group Gal(κ/κ)). The right side of (126.96.36.199) is visibly a product of matrix algebras
over factor fields of Z =
Zv, so D splits at all -adic places of Z. Writing
d2 = [D : Z], the Zv-algebra isomorphism Dv EndZv (Vv(A)) for v| implies that
Vv(A) has Zv-dimension d for all such v, so V (A) is free of rank d over Z . Hence,
[Z : Q ]d = 2g, so g = (1/2)[Z : Q] [D : Z]. This is the asserted dimension
formula in case (3), and the proof also applies in cases (1) and (2) (as will be used
below). The formula for invv(D) in case (3) with v|p is proved in A.1.3, resting
on preliminary work in A.1.1 and A.1.2, and that proof is applicable regardless of
whether Z is CM or totally real. This completes the proof of case (3), and in cases
(1) and (2) (so
= q) it establishes the formula invv(D) = [Zv : Qp]/2 mod Z for
p-adic places v of Z.
Consider case (1) (equivalently, π =
with n even), so D is a central
division algebra over Z = Q split away from p and ∞ with invp(D) = 1/2 mod Z.
This forces inv∞(D) = 1/2 mod Z, so D is a quaternion division algebra over Q.
In particular, the dimension formula yields g = 1, so A is an elliptic curve. In view
of the other possibilities for Z, these are the only cases for which D is a central
quaternion division algebra over Q. The elliptic curves E that arise in such cases
must be supersingular (since it is classical that
is commutative in the or-
dinary case). Moreover, since it is classical that
is a quaternion division
algebra in the supersingular case, it follows for Q-dimension reasons that the injec-
tion D =
is an equality. In other words, all endomorphisms
of Eκ are defined over κ. This settles case (1).
Finally, consider case (2). Since the numbers
with odd n are Galois
conjugate over Q, there is exactly one isogeny class that arises in this case. For
the unique p-adic place v of Z = Q(
p), the formula for invv(D) vanishes. Hence,
D splits away from the two real places of Z, so its order in Br(Z) divides 2 and
the dimension formula says g = [D : Z]. Thus, either D = Z and A is an elliptic
curve or D is the unique central quaternion division algebra over Z split away from
the real places and A is an abelian surface. The first case cannot happen, since
otherwise the quadratic field Z would provide a CM structure on the elliptic curve,
contradicting Proposition 188.8.131.52(2) since D = Z is a real quadratic field.