1.5. CM TYPES 65

1.5. CM types

Let A be an isotypic abelian variety of dimension g 0 over a field K such

that A admits suﬃciently many complex multiplications. By Theorem 1.3.4, we

may and do choose a CM field L ⊂

End0(A)

with degree 2g. (Conversely, by

Theorem 1.3.1.1, the existence of such an L forces A to be isotypic.) It turns out

that the L-linear isogeny class of A is encoded in terms of a rather simple discrete

invariant when char(K) = 0 and K is algebraically closed. We wish to review the

basic features of this invariant, called the CM type, and to discuss some useful

replacements for it in positive characteristic.

The order O = L ∩ End(A) in L acts on A over K, and is called the CM order.

It acts K-linearly on the tangent space T = Lie(A) at the origin, so if char(K) = 0

then L = O ⊗Z Q acts K-linearly on T , whereas if char(K) = p 0 then O/(p)

acts K-linearly on T . In particular, if char(K) = 0 then T is an L ⊗Q K-module

whose isomorphism class is an invariant of the L-linear isogeny class of A over K;

nothing of the sort is true when char(K) = p 0.

1.5.1. Characteristic 0. We now focus on the case char(K) = 0. Let K /K

be an algebraically closed extension. Since L ⊗Q K Ki for finite (separable)

extensions Ki/K, any L ⊗Q K-module M canonically decomposes as Mi for

a Ki-vector space Mi. Thus, if dimK M is finite then the isomorphism class of

M is determined by the numbers dimKi Mi, which in turn are determined by the

isomorphism class of the L ⊗Q K -module M ⊗K K .

The K -algebra L ⊗Q K has a very simple form: it is

ϕ

Kϕ where ϕ ranges

through all field embeddings L → K and Kϕ denotes K viewed as an L-algebra

via ϕ. Hence, any L ⊗Q K -module M decomposes into a corresponding product

of eigenspaces Mϕ over K on which L acts through ϕ. We conclude that for

an L ⊗Q K-module M with finite K-dimension, the isomorphism class of M is

determined by the numbers dimK (M ⊗K K )ϕ as ϕ varies through Hom(L, K ).

On the set Hom(L, K ) = Hom(L, Q) (with Q the algebraic closure of Q in K )

there is a natural involution defined by precomposition with the intrinsic complex

conjugation ι of the CM field L (i.e., the non-trivial automorphism of L over its

maximal totally real subfield L+). This decomposes the set Hom(L, K ) of size

2g into g “conjugate pairs” of embeddings. In the special case K = C we can

also compute the involution on Hom(L, K ) by using composition with complex

conjugation on K = C.

An especially interesting example is the L ⊗Q K -module M = T ⊗K K with

T = Lie(A) for a CM abelian variety A over K with complex multiplication by L.

There is a non-trivial constraint on the eigenspaces (T ⊗K K )ϕ for the L-action

on T ⊗K K (with ϕ varying through the embeddings ϕ : L → K ):

1.5.2. Lemma. When char(K) = 0, each eigenspace (T ⊗K K )ϕ is at most 1-

dimensional over K . If Φ denotes the set of g distinct embeddings ϕ : L → K

for which there is a ϕ-eigenline in T ⊗K K then Φ contains no “conjugate pairs”.

That is, we have a disjoint union decomposition Hom(L, K ) = Φ (Φ ◦ ι).

Proof. By considerations with direct limits (as in the proof of Proposition

1.2.6.1), we may and do first arrange that K is finitely generated over Q. The

choice of algebraically closed extension K /K does not matter, so we can replace