2 INTRODUCTION

the structure of the endomorphism algebra

End0(A)

(working in the isogeny cate-

gory over κ) in terms of the Weil q-integer of A, with q = #κ; see [121, Thm. 1].

It follows from Tate’s work (see 1.6.2.5) that an abelian variety A over a finite field

κ admits suﬃciently many complex multiplications in the sense that its endomor-

phism algebra

End0(A)

contains a CM

subalgebra3

L of rank 2 dim(A). We will

call such an abelian variety (in any characteristic) a CM abelian variety and the

embedding L →

End0(A)

a CM structure on A.

Grothendieck showed that over any algebraically closed field K, an abelian

variety that admits suﬃciently many complex multiplications is isogenous to an

abelian variety defined over a finite extension of the prime field [89]. This was

previously known in characteristic zero (by Shimura and Taniyama), and in that

case there is a number field K ⊂ K such that the abelian variety can be defined

over K (in the sense of 1.7.1). However in positive characteristic such abelian

varieties can fail to be defined over a finite subfield of K; examples exist in every

dimension 1 (see Example 1.7.1.2).

Abelian varieties in mixed characteristic. In characteristic zero, an abelian

variety A gives a representation of the endomorphism algebra D =

End0(A)

on the

Lie algebra Lie(A) of A. If A has complex multiplication by a CM algebra L of

degree 2 dim(A) then the isomorphism class of the representation of L on Lie(A) is

called the CM type of the CM structure L →

End0(A)

on A (see Lemma 1.5.2 and

Definition 1.5.2.1).

As we noted above, every abelian variety over a finite field is a CM abelian vari-

ety. Thus, it is natural to ask whether every abelian variety over a finite field can be

“CM lifted” to characteristic zero (in various senses that are made precise in 1.8.5).

One of the obstacles4 in this question is that in characteristic zero there is the no-

tion of CM type that is invariant under isogenies, whereas in positive characteristic

whatever can be defined in an analogous way is not invariant under isogenies. For

this reason we will use the terminology “CM type” only in characteristic zero.

For instance, the action of the endomorphism ring R = End(A0) of an abelian

variety A0 on the Lie algebra of A0 in characteristic p 0 defines a representa-

tion of R/pR on Lie(A0). Given an isogeny f : A0 → B0 we get an identification

End0(A0)

=

End0(B0)

of endomorphism algebras, but even if End(A0) = End(B0)

under this identification, the representations of this common endomorphism ring

on Lie(A0) and Lie(B0) may well be non-isomorphic since Lie(f) may not be an

isomorphism. Moreover, if we have a lifting A of A0 over a local domain of char-

acteristic 0, in general the inclusion End(A) ⊂ End(A0) is not an equality. If the

inclusion

End0(A)

⊂

End0(A0)

is an equality then the character of the representa-

tion of End(A0) on Lie(A0) is the reduction of the character of the representation of

End(A) on Lie(A). This relation can be viewed as an obstruction to the existence

of CM lifting with the full ring of integers of a CM algebra operating on the lift;

see 4.1.2, especially 4.1.2.3–4.1.2.4, for an illustration.

In the case when End(A0) contains the ring of integers OL of a CM algebra

L ⊂

End0(A0)

with [L : Q] = 2 dim(A0), the representation of OL/pOL on Lie(A0)

turns out to be quite useful, despite the fact that it is not an isogeny invariant. Its

class in a suitable K-group will be called the Lie type of (A0, OL → End(A0)).

3A

CM algebra is a finite product of CM fields; see Definition 1.3.3.1.

4surely

also part of the attraction