INTRODUCTION 3

The above discrepancy between the theories in characteristic zero and charac-

teristic p 0 is the basic phenomenon underlying this entire book. Before dis-

cussing its content, we recall the following theorem of Honda and Tate ([50, §2,

Thm. 1] and [121, Thm. 2]).

For an abelian variety A0 over a finite field κ there is a finite extension

κ of κ and an isogeny (A0)κ → B0 such that B0 admits a CM lifting

over a local domain of characteristic zero with residue field κ .

This result has been used in the study of Shimura varieties, for settings where the

ground field is an algebraic closure of Fp and isogeny classes (of structured abelian

varieties) are the objects of interest; see [135]. Our starting point comes from the

following questions which focus on controlling ground field extensions and isogenies.

For an abelian variety A0 over a finite field κ, to ensure the existence

of a CM lifting over a local domain with characteristic zero and residue

field κ of finite degree over κ,

(a) may we choose κ = κ?

(b) is an isogeny (A0)κ → B0 necessary?

These questions are formulated in various precise forms in 1.8.

An isogeny is necessary. Question (b) was answered in 1992 (see [93]) as follows.

There exist (many) abelian varieties over Fp that do not admit any CM

lifting to characteristic zero.

The main point of [93] is that a CM liftable abelian variety over Fp can be defined

over a small finite field. This idea is further pursued in Chapter 3, where the size,

or more accurately the

minima5

of the size, of all possible fields of definition of the

p-divisible group of a given abelian variety over Fp is turned into an obstruction for

the existence of a CM lifting to characteristic 0. This is used to show (in 3.8.3) that

in “most” isogeny classes of non-ordinary abelian varieties of dimension 2 over

finite fields there is a member that has no CM lift to characteristic 0. (In dimension

1 a CM lift to characteristic 0 always exists, over the valuation ring of the minimal

possible p-adic field, by Deuring Lifting Theorem; see 1.7.4.6.) We also provide

effectively computable examples of abelian varieties over explicit finite fields such

that there is no CM lift to characteristic 0.

A field extension might be necessary—depending on what you ask.

Bearing in mind the necessity to modify a given abelian variety over a finite field to

guarantee the existence of a CM lifting, we rephrase question (a) in a more precise

version (a) below.

(a) Given an abelian variety A0 over a finite field κ of characteris-

tic p, is it necessary to extend scalars to a strictly larger finite field

κ ⊃ κ (depending on A0) to ensure the existence of a κ -rational isogeny

(A0)κ → B0 such that B0 admits a CM lifting over a characteristic 0

local domain R with residue field κ ?

It turns out there are two quite different answers to question (a) , depending on

whether one requires the local domain R of characteristic 0 to be normal. The

subtle distinction between using normal or general local domains for the lifting

5The

size of a finite field κ1 is smaller than the size of a finite field κ2 if κ1 is isomorphic to

a subfield of κ2, or equivalently if #κ1 | #κ2. Among the sizes of a family of finite fields there

may not be a unique minimal element.