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

π2

= q =

pn,

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

(1.6.2.1) 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 (1.6.2.1) is visibly a product of matrix algebras

over factor fields of Z =

v|

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

π2

= q) it establishes the formula invv(D) = [Zv : Qp]/2 mod Z for

p-adic places v of Z.

Consider case (1) (equivalently, π =

±pn/2

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

End0(Eκ)

is commutative in the or-

dinary case). Moreover, since it is classical that

End0(Eκ)

is a quaternion division

algebra in the supersingular case, it follows for Q-dimension reasons that the injec-

tion D =

End0(E)

→

End0(Eκ)

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

±pn/2

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 1.3.6.4(2) since D = Z is a real quadratic field.