Fix a prime = p. Since S is connected and A [
is finite ´ etale over S, an
S-endomorphism of A [
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
injective for all s S. We can therefore speak of an element of
over K in the sense that it is the image of a unique element of D =
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
with [L : Q] = 2g =
2 dim(As), so each As is isotypic. By Theorem, L is its own centralizer in
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
is central,
so it centralizes L and hence must lie in the image of L. In particular, ϕs lifts to
an element of
= D that is necessarily central (as we may compute after
applying the injective specialization map D
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, under any embedding
ι : Q[ϕs] C we have each ι(ϕs)ι(ϕs) = qs for qs = #κ(s)
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(

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 [
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
(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
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