1.8. CM LIFTING QUESTIONS 87

action by i on the tangent line Lie(EQ(i)) = Lie(E) ⊗R Q(i) would preserve the

R-submodule Lie(E). But Lie(E) is a free R-module of rank 1 since R is local

and E is R-smooth, so the i-action on this R-module is multiplication by some

element r ∈ R. By working over Q(i) we have seen that we get the multiplier i, so

necessarily r = i. Since i ∈ R due to the definition of R, we have a contradiction.

In Example 1.8.3, the base ring R is not normal. This is essential, since in the

normal case there is no obstruction to extending maps between abelian schemes:

1.8.4. Lemma. For a normal domain R with fraction field K, the functor A

AK from abelian schemes over R to abelian varieties over K is fully faithful.

Proof. This is a special case of a general lemma of Faltings [36, §2, Lemma 1]

concerning homomorphisms between semi-abelian schemes over a normal scheme

(the proof of which simplifies considerably in the case of abelian schemes).

1.8.4.1. For normal R, Lemma 1.8.4 provides a specialization map

End0(AK)

=

End0(A)

:= Q ⊗Z End(A) →

End0(Aκ)

between endomorphism algebras, and likewise for endomorphism rings. This makes

normality a natural property to impose on R when studying questions about CM

lifts. If R is not normal then

End0(AK)

may be larger than

End0(A),

so it is not

evident how to compare endomorphism algebras of the K-fiber and κ-fiber.

Hence, for general R we just work with the specialization map of endomorphism

algebras

End0(A)

→

End0(Aκ).

This map can fail to be surjective. An elementary

example is an elliptic curve over Z(p) for a prime p (since elliptic curves over finite

fields always admit complex multiplication, by Corollary 1.6.2.5, whereas elliptic

curves over Q have endomorphism algebra Q). In contrast with Lemma 1.8.2, the

specialization map of endomorphism rings End(A) → End(Aκ) can have cokernel

that is not torsion-free, even when R is normal. (In Chapter 4 we will see many

natural examples of this phenomenon in our study of CM lifting problems, when

we consider lifting questions for specific orders in CM fields; e.g., see 4.1.2.)

1.8.4.2. Remark. In the setting of Lemma 1.8.4, if λ : A → B is a homomorphism

between abelian schemes over R and λK is an isogeny then λ is an isogeny (i.e., λ is

fiberwise surjective with finite kernel; see §3.3 for a general discussion of isogenies

for abelian schemes). To prove this, choose a K-homomorphism μK : BK → AK

such that μK ◦ λK is multiplication by a non-zero integer n. The homomorphism

μ : B → A extending μK therefore satisfies μ ◦ λ = [n]A, so λ has a fiberwise finite

kernel and hence is an isogeny by fibral dimension considerations.

1.8.5. CM lifting problems. To formulate the lifting questions that we study in

subsequent chapters, let Fq be a finite field of size q and let B be an abelian variety

of dimension g 0 over Fq. Assume B is isotypic over Fq (necessary and suﬃcient

for B to admit a structure of CM abelian variety with complex multiplication

by a CM field, by Theorem 1.3.1.1 and Corollary 1.6.2.5). Let Bκ denote the

scalar extension of B over a finite extension field κ/Fq. Consider the following five

assertions concerning the existence of a CM lifting of B or Bκ to characteristic 0.

• (CML) CM lifting: there is a local domain R with characteristic 0 and residue

field Fq, an abelian scheme A over R with relative dimension g equipped with