88 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

a CM field L ⊂

End0(A)

:= Q ⊗Z End(A) satisfying [L : Q] = 2g, and an

isomorphism φ : AFq B as abelian varieties over Fq.

• (R) CM lifting after finite residue field extension: there is a local domain R

with characteristic 0 and residue field κ of finite degree over Fq, an abelian

scheme A over R with relative dimension g equipped with an action (in the

isogeny category over R) by a CM field L with [L : Q] = 2g, and an isomor-

phism φ : Aκ B ×Spec(Fq) Spec(κ) as abelian varieties over κ.

• (I) CM lifting up to isogeny: there is a local domain R with characteristic 0

and residue field Fq, an abelian scheme A over R with relative dimension g

equipped with an action (in the isogeny category over R) by a CM field L

with [L : Q] = 2g, and an isogeny AFq → B of abelian varieties over Fq.

• (IN) CM lifting to normal domains up to isogeny: there is a normal local

domain R with characteristic 0 and residue field Fq such that (I) is satisfied

for B using R.

• (RIN) CM lifting to normal domains up to isogeny after finite residue field

extension: there is a normal local domain R with characteristic 0 and residue

field κ of finite degree over Fq such that (R) is satisfied for B using R except

that φ is only required to be an isogeny rather than an isomorphism.

• (sCML) strong CM lifting: For every CM field L ⊂

End0(B)

with [L : Q] = 2g

such that OL ⊂ End(B), the abelian variety B satisfies (CML) using a lifting

A such that the Q-subalgebra

End0(A)

⊂

End0(B)

contains L.

1.8.5.1. Remark. By expressing a local ring as a direct limit of local subrings

essentially of finite type over Z, in the formulation of (R) there is no loss of generality

in replacing κ with an algebraic closure of Fq or allowing κ to vary over all extensions

of Fq. Likewise, the normality condition in (RIN) is irrelevant because it can be

attained at the cost of a finite residue field extension (by a specialization argument

that we will give in 2.1.1), and in (IN) we can assume R is complete since essentially

finite type Z-algebras are excellent (ensuring that normality is preserved under

completion of such rings along an ideal). Even in (I) we can assume R is complete

local noetherian since we may first descend to a local noetherian domain R0 ⊂ R

of characteristic 0 with residue field Fq, and then note that the completion R0 of

R0 has a minimal prime of residue characteristic 0 (as R0 → R0 is faithfully flat).

By Remark 1.6.5.2, (RIN) has an aﬃrmative answer for any isotypic B over

Fq, and the CM lift can be chosen using any CM maximal commutative subfield

L ⊂

End0(B).

There are several refinements we wish to answer:

(1) Is a residue field extension necessary? That is, does (IN) hold for every B?

(2) If (IN) does not hold for every B, can we characterize when it holds? And

how about (I) in general (i.e., drop normality, but permit an isogeny without

increasing the residue field)?

(3) Is an isogeny necessary? That is, does (R) hold for every B (requiring the

local domain R to be normal is not a constraint, since we are allowing a finite

extension on κ; cf. Remark 1.8.5.1), or does even (CML) hold for every B?

These questions can be made more specific in several respects. For example,

since the Q-simple

End0(B)

is usually non-commutative, it generally contains more

than one CM maximal commutative subfield L (up to conjugacy) and so we can

pose the CM lifting questions requiring an order in a particular choice of L to lift