60 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

A0[p∞] t

with the Cartier dual of

A0[p∞],

the infinitesimal deformation theory of

A0 equipped with this extra structure is the same as that of its p-divisible group

equipped with the analogous induced extra structure.

Consider a homomorphism φ0 : A0 → A0

t

and the associated homomorphism

f0 :

A0[p∞]

→

A0[p∞] t A0[p∞]t.

Clearly φ0 is an isogeny if and only if f0 is an isogeny, and we saw in 1.4.3.4 that

φ0 is symmetric with respect to double duality of abelian varieties if and only if f0

is skew-symmetric (i.e., f0

t

= −f0) with respect to the canonical isomorphism of a

p-divisible group with its Serre double dual.

This leads us to define a quasi-polarization of a p-divisible group G over a

complete local noetherian ring to be a skew-symmetric homomorphism f : G →

Gt

that induces an isogeny between the special fibers. (We will see in 3.3.8 that it

is equivalent to say the skew-symmetric f is an isogeny in the sense of 3.3.5, so

we can thereby define the notion of quasi-polarization over any base scheme.) The

ampleness aspect of a polarization cannot be encoded in terms of p-divisible groups,

but quasi-polarizations are nonetheless a helpful concept when using p-divisible

groups to study abelian varieties and their deformations, as the following example

illustrates.

1.4.5.4. Example. As a special case of the Serre-Tate deformation theorem, if

R is a complete local noetherian ring of residue characteristic p 0 and A0 is an

abelian variety over the residue field, then a deformation of A0[p∞] to a p-divisible

group G over R corresponds to a deformation of A0 to a formal abelian scheme A

over R. If A0 is equipped with a CM structure and we demand that this structure

lifts to A (via the injection End(A) → End(A0)) then A can fail to be algebraic (i.e.,

it may not be the formal completion of a proper R-scheme). Explicit CM examples

of this type are given in 4.1.2; also see the discussion immediately following the

statement of Theorem 2.2.3.

To ensure algebraicity of A, we need to encode the deformation of a polariza-

tion. More specifically, choose a polarization φ0 : A0 → A0

t

and suppose that the

corresponding map f :

A0[p∞]

→

A0[p∞] t

=

A0[p∞]t

lifts to R (as can happen in

at most one way, by Proposition 1.4.4.3). Let φ : A →

At

be the corresponding

unique homomorphism that lifts φ0 (in accordance with the Serre–Tate deforma-

tion theorem). If P denotes the formal Poincar´ e bundle on A ×

At

(which lifts the

Poincar´ e bundle P0 on A0 × A0)

t

then

(1,φ)∗P

is a line bundle on A lifting the line

bundle

(1,φ0)∗P0

on A0 that is ample (due to φ0 being a polarization). Hence,

by Grothendieck’s algebraization theorems [34, III1, 5.4.1, 5.4.5], in such cases A

is algebraic, so it arises from a unique abelian R-scheme A deforming A0.

There is a special case in which the liftability of all polarizations comes “for

free”: the Serre–Tate canonical lifting of an ordinary abelian variety A0 over a

perfect field k of characteristic p 0. To explain these concepts, we first note that

by the perfectness of k, the connected-´ etale sequence of every p-divisible group X

over k is split (exactly as for finite commutative k-groups), so X is (uniquely) the

product of an ´ etale p-divisible group and a connected p-divisible group; see [87,

I.2]. Letting g = dim(A0), since an ´ etale p-divisible group over k has connected

Serre dual (as we may check over k) and

A0[p∞]

=

A0[p∞]0

×

A0[p∞]´ et

is of height

2g yet isogenous to its Serre dual

A0[p∞]t A0[p∞] t

(as A0 is isogenous to A0),

t

we

see that the p-rank of

A0[p∞]

(i.e., height of

A0[p∞]´ et

) is at most g. We say that A0