14 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

scheme amounts to a group scheme diagram for a smooth proper R-scheme having

geometrically connected fibers, the abelian scheme A over R descends to an abelian

scheme over Ri0 for some suﬃciently large i0.

The residue field κi0 of Ri0 is merely a subfield of κ. By [34, 0III, 10.3.1],

there is a faithfully flat local extension Ri0 → R with R noetherian and having

residue field κ over κi0 . By faithful flatness, every minimal prime of R has residue

characteristic 0, so we can replace R with its quotient by such a prime to obtain a

solution over a complete local noetherian domain with residue field κ.

Typically our liftings will be equipped with additional structure such as a po-

larization, and so the existence of an aﬃrmative solution for our lifting problem

(for a given A0) often amounts to an appropriate deformation ring R for A0 (over a

Cohen ring for κ) admitting a generic point in characteristic 0; the coordinate ring

of the corresponding irreducible component of Spec(R) is such an R. If κ → κ is

an extension of fields and W → W is the associated extension of Cohen rings then

often there is a natural isomorphism R W ⊗W R relating the corresponding de-

formation rings for A0 and (A0)κ (see 1.4.4.5, 1.4.4.13, and 1.4.4.14). Thus, if R

has a generic point of characteristic 0 then so does R. Hence, to prove an aﬃrmative

answer to lifting questions as above it is usually enough to consider algebraically

closed κ. For example, the general lifting problem for polarized abelian varieties

(allowing polarizations for which the associated symmetric isogeny A0 → A0

t

is not

separable) was solved aﬃrmatively by Norman-Oort [85, Cor. 3.2] when κ = κ, and

the general case follows by deformation theory (via 1.4.4.14 with O = Z).

1.1.2. Refinements. When a lifting problem as above has an aﬃrmative solution,

it is natural to ask if the (complete local noetherian) base ring R for the lifting

can be chosen to satisfy nice ring-theoretic properties, such as being normal or a

discrete valuation ring. Slicing methods allow one to find an R with dim(R) = 1

(see 2.1.1 for this argument), but normalization generally increases the residue field.

Hence, asking that the complete local noetherian domain R be normal or a discrete

valuation ring with a specified residue field κ is a non-trivial condition unless κ is

algebraically closed.

We are interested in versions of the lifting problem for finite κ when we lift not

only the abelian variety but also a large commutative subring of its endomorphism

algebra. To avoid counterexamples it is sometimes necessary to weaken the lifting

problem by permitting the initial abelian variety A0 to be replaced with another

in the same isogeny class over κ. In 1.8 we will precisely formulate several such

lifting problems involving complex multiplication, and the main result of our work

is a rather satisfactory solution to these lifting problems.

1.1.3. Purpose of this chapter. Much of the literature on complex multiplica-

tion involves either (i) working over an algebraically closed ground field, (ii) making

unspecified finite extensions of the ground field, or (iii) restricting attention to sim-

ple abelian varieties. To avoid any uncertainty about the degree of generality in

which various foundational results in the theory are valid, as well as to provide a

convenient reference for subsequent considerations, in this chapter we provide an

extensive review of the algebraic theory of complex multiplication over a general

base field. This includes special features of the theory over finite fields and over

fields of characteristic 0, and for some important proofs we refer to the original

literature (e.g., papers of Tate). Some arithmetic aspects (such as reflex fields and