44 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Now set M =
M∗(G)[1/p],
whose K0-dimension is h. This is equipped with
the required Frobenius-semilinear injective endomorphism (that is moreover F -
linear by functoriality), so it is free of some rank ρ 1 over K0 ⊗Qp F and hence
h = [F : Qp]ρ. It follows that [F : Qp] h, with equality if and only if ρ = 1. This
proves (1).
Finally, assuming we are in this rank-1 case, it remains to prove that F is its
own centralizer in
End0(G).
To compute the centralizer of F , first observe that
the Dieudonn´ e module functor (on the isogeny category) is valued in the category
of K0-vector spaces, so every element f
End0(G)
that commutes with the F -
action induces a K0 ⊗Qp F -linear endomorphism of
M∗(G)[1/p].
We know that
M∗(G)[1/p]
is free of rank 1 over K0 ⊗Qp F , so
M∗(f)
acts as multiplication by
some c K0 ⊗Qp F . Since
M∗(f)
also commutes with the action of F that is
semilinear over the absolute Frobenius σ of K0, we have 1)(c) = c. This forces
c F , as desired.
1.4.4. Deformation theory. Let R be a local ring with residue field κ. The func-
tor A from abelian schemes over R to abelian varieties over κ is faithful. This
follows from two facts: the collection of finite ´ etale subgroup schemes A[N] for N
not divisible by char(κ) is schematically dense in A (due to the fiberwise denseness
and [34, IV3, 11.10.9]), and passage to the special fiber is faithful on finite ´etale
R-schemes. When considering deformation problems for abelian varieties equipped
with endomorphisms or a polarization (viewed as a special kind of isogeny), this
faithfulness result is implicitly used without comment.
1.4.4.1. Remark. For abelian R-schemes A and B, the injective reduction map
Hom0(A,
B) := Q ⊗Z Hom(A, B)
Hom0(Aκ,Bκ)
gives meaning to the intersection
Hom0(A,
B) Hom(Aκ,Bκ). This intersection
contains Hom(A, B) but can be strictly larger. To make an example, let R be a
discrete valuation ring with fraction field K of characteristic 0 and residue field κ
of characteristic p 0, and let E be an elliptic curve over R such that E[p]K is
constant and
is ordinary. There are p + 1 cyclic subgroups of E[p]K with order
p, so via R-flat closure there are p + 1 closed R-flat subgroups C E of order p.
For any local extension of discrete valuation rings R R , the R -subgroups {CR }
of ER exhaust the p + 1 possibilities over R . Due to the connected-´ etale sequence
over R, it follows that exactly one such C E is connected, so the p others are ´etale
and hence have reduction equal to the same (unique) ´ etale κ-subgroup of Eκ[p].
If C, C E[p] are distinct ´ etale subgroups of order p then the kernels of the
isogenies f : E E/C and f : E E/C are distinct over K but the same over
κ. Since the reductions of f and f have the same kernel, in the isogeny category
of elliptic curves over R the element f (f
)−1

Hom0(E/C
, E/C) has reduction
that is a morphism of elliptic curves (and even an isomorphism, with inverse given
by the reduction of f f
−1).
If f f
−1
were a morphism of elliptic curves over R
then it would have to be an isomorphism (since its reduction is an isomorphism),
and so C would be in the orbit of C under Aut(E) = Aut(EK). These orbits
have size at most #Aut(EK)/2 3, so we can find C and C not in the same
Aut(E)-orbit whenever j(EK) = 0, 1728 or p 3.
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