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
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