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
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
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