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 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 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
Previous Page Next Page