1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 33

Fix a prime = p. Since S is connected and A [

n]

is finite ´ etale over S, an

S-endomorphism of A [

n]

is uniquely determined by its effect on a single geometric

fiber over S. But maps between abelian varieties are uniquely determined by their

effect on -adic Tate modules when is a unit in the base field, so we conclude (via

consideration of -power torsion) that the specialization map D →

End0(As)

is

injective for all s ∈ S. We can therefore speak of an element of

End0(As)

“lifting”

over K in the sense that it is the image of a unique element of D =

End0(A)

under

the specialization mapping at s. This will be of interest when s is a closed point

and we consider the qs-Frobenius endomorphism of As over the finite residue field

κ(s) at s (with qs = #κ(s)).

By Theorem 1.3.4, we can choose a CM field L ⊂ D with [L : Q] = 2g. In

particular, for each s ∈ S the field L embeds into

End0(As)

with [L : Q] = 2g =

2 dim(As), so each As is isotypic. By Theorem 1.3.1.1, L is its own centralizer in

End0(As).

Take s to be a closed point of S, and let qs denote the size of the finite

residue field κ(s) at s. The qs-Frobenius endomorphism ϕs ∈

End0(As)

is central,

so it centralizes L and hence must lie in the image of L. In particular, ϕs lifts to

an element of

End0(A)

= D that is necessarily central (as we may compute after

applying the injective specialization map D →

End0(As)).

That is, ϕs ∈ Z ⊂ D

for all closed points s ∈ S.

Let Z be the subfield of Z generated over Q by the lifts of the endomorphisms

ϕs as s varies through all closed points of S. Each Q[ϕs] is a totally real field

since Z is totally real. By Weil’s Riemann Hypothesis for abelian varieties over

finite fields (see the discussion following Definition 1.6.1.2), under any embedding

ι : Q[ϕs] → C we have each ι(ϕs)ι(ϕs) = qs for qs = #κ(s) ∈

pZ,

so the real number

ι(ϕs) is a power of

√

p. Hence, the subfield Q[ϕs] ⊂ Z is either Q or Q(

√

p), so the

subfield Z ⊂ Z is either Q or Q(

√

p).

Let η be a geometric generic point of S, and let Γ be the associated absolute

Galois group for the function field of S. Because each A [

n]

is finite ´ etale over S, the

representation of Γ on V (A) factors through the quotient π1(S, η). The Chebotarev

Density Theorem for π1(S, η) [97, App. B.9] says that the Frobenius elements at the

closed points of S generate a dense subgroup of the quotient π1(S, η) of Γ. Thus,

the image of Q [Γ] in EndQ (V (A)) is equal to the subalgebra Z := Q ⊗Q Z

generated by the endomorphisms ϕs. We therefore have an injective map

Q ⊗Q D → EndQ

[Γ]

(V (A)) = EndZ (V (A)).

By Zarhin’s theorem [134] (see [80, XII, §2] for a proof valid for all p, especially

allowing p = 2) this injection is an isomorphism, so we conclude that Z is central

in EndZ (V (A)). But the center of this latter matrix algebra is Z , so the inclusion

Z ⊂ Z is an equality. Hence, the inclusion Z ⊂ Z is an equality as well. Since

Z is either Q or Q(

√

p), we are done.

1.4. Dieudonn´ e theory, p-divisible groups, and deformations

To solve problems involving lifts from characteristic p to characteristic 0, we

need a technique for handling p-torsion phenomena in characteristic p 0. The

two main tools for this purpose in what we shall do are Dieudonn´ e theory and

p-divisible groups. For the convenience of the reader we review the basic facts in

this direction, and for additional details we refer to [119], [110, §6], and [75] for