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 (σ ⊗ 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 Aκ 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 Eκ

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.