1.5. CM TYPES 69

For an algebraically closed extension K /K we can consider the eigenspace

decomposition of T ⊗K K over κ ⊗Fp K =

ϕ

Kϕ where ϕ ranges over the 2g

distinct embeddings of κ into K . This could fail to resemble the CM types that

arise in characteristic 0 because (as we shall see in later examples, such as in Remark

2.3.4) there may be conjugate pairs occurring among the ϕ for which T ⊗K K has

a non-zero ϕ-eigenspace with respect to its K -linear κ-action.

In such cases, the composite action

O → End(A) → End(A)/(p) → EndK (T )

does not “look like the reduction of a CM type”, and so this provides an obstruction

for A equipped with its O-action to lift to characteristic 0. There is no dimension

obstruction to such lifting: each ϕ-eigenspace in T ⊗K K = Lie(AK ) has K -

dimension at most 1. To prove this, first note that the Dieudonn´ e module D :=

M∗(AK

[p∞])

is free of rank 1 over

O ⊗Z W (K ) = O(p) ⊗Z(p) W (K ) = OL ⊗Z W (K )

by Proposition 1.4.3.9(2) (or Proposition 1.2.5.1 and W (K )-rank considerations),

so D/pD is free of rank 1 over κ ⊗Fp K . The formal group AK is the identity

component of the p-divisible group AK [p∞] (Example 1.4.3.6), and its tangent

space coincides with that of AK . Hence, by 1.4.3.2(4),

T ⊗K K Lie(AK

[p∞]) (D/F(D))∨,

where F : D → D denotes the semilinear Frobenius endomorphism. By naturality,

this composite isomorphism is κ⊗Fp K -linear, so T ⊗K K is monogenic over κ⊗Fp K

since the K -linear dual of a monogenic κ⊗Fp K -module is monogenic (as κ⊗Fp K

is a finite ´ etale K -algebra). Each ϕ-eigenspace of T ⊗K K is therefore monogenic

over K , which is to say is of dimension at most 1 over K .

To go beyond Example 1.5.5.2, an obstruction to the existence of a CM lift over

a normal local domain of characteristic 0 will be formulated precisely later (see 2.1.5

and 4.1.2). This will be used to exhibit examples (e.g., in 4.1.2) of abelian varieties

over finite fields for which there is no such lift. Such examples are interesting due

to Corollary 1.6.2.5 below, according to which every abelian variety over a finite

field admits suﬃciently many complex multiplications.

Although the Lie algebra fails to be an isogeny invariant for the study of CM

abelian varieties in positive characteristic (and

End0(A)

does not act on the tangent

space when char(K) 0), there is an alternative linear object attached to a CM

abelian variety A that serves as a good substitute when char(K) = p 0: the

p-divisible group A[p∞], or its (contravariant) Dieudonn´ e module

M∗(A[p∞])

when

K is perfect.

Letting B = A in Proposition 1.2.5.1, we see that Zp ⊗Z End(A) acts faithfully

on

A[p∞].

Hence, Qp ⊗Q

End0(A)

acts faithfully on

A[p∞]

in the isogeny category

of p-divisible groups over K. In particular, if K is perfect (e.g., finite) and A is

an isotypic CM abelian variety over K with complex multiplication by the CM

field L (see Theorem 1.3.4) then Lp := Qp ⊗Q L acts faithfully and linearly on the

vector space

M∗(A[p∞])[1/p]

of rank 2g over the absolutely unramified p-adic field

W (K)[1/p].

This W (K)[1/p]-linear faithful Lp-action for perfect K with char(K) = p is

an analogue of a classical construction when char(K) = 0: the action of L on the