1.6. ABELIAN VARIETIES OVER FINITE FIELDS 75

elliptic curve E, so Gκ αp as κ-groups (since Gκ is local-local of order p, so

M∗(Gκ)

= κ with F = V = 0). But OK,(p) is an absolutely unramified discrete

valuation ring, so there are no finite flat group schemes over OK,(p) with special

fiber αp (by the classification results in [123]). This contradiction shows that E

and E are not isogenous (so AK is not isotypic), as claimed. In fact, we have

proved something stronger: if Kp denotes the p-adic completion of K then AKp is

not isotypic.

1.6.5. CM lifting after a field extension and isogeny. The proof of the sur-

jectivity aspect of the Honda-Tate theorem requires constructing abelian varieties

having prescribed properties over finite fields. The idea is to relate simple abelian

varieties over finite fields to simple factors of reductions of CM abelian varieties

over number fields, at least after some finite extension on the initial finite field.

(See [50] or [121, Lemme 3] for details.)

One can ask (as Honda implicitly did at the end of [50, §2]) to do better by

arranging simplicity to hold for the reduction of a CM abelian variety over a number

field (thereby eliminating the need to pass to a simple factor). Building on earlier

work of Honda, such an improved lifting theorem was proved by Tate [121, Thm. 2]

(and is really the starting point for the many lifting questions about CM abelian

varieties that we consider in this book):

1.6.5.1. Theorem (Honda, Tate). For any isotypic abelian variety A over a finite

field κ, there is a finite extension κ /κ such that Aκ is isogenous to the reduction

of a CM abelian variety with good reduction over a p-adic field with residue field κ .

Proof. By Corollary 1.6.2.5, there is a CM field L ⊂

End0(A)

with [L : Q] =

2 dim(A). The field L is its own centralizer in

End0(A),

so it contains an element π

which acts by the q-Frobenius endomorphism on A, where q = #κ. Let g = dim(A).

Since A is κ-isotypic, Tate’s work on isogenies among abelian varieties over finite

fields [118] gives two results for A: (i) the common characteristic polynomial over Q

for the action of π on the Tate modules of A is a power of an irreducible polynomial

fπ over Q (necessarily the minimal polynomial of π over Q), and (ii) A is κ-isogenous

to any g-dimensional isotypic abelian variety over κ whose q-Frobenius is a zero of

fπ. Moreover, these properties persist after replacing κ with any finite extension

κ (and replacing π with π[κ :κ]), due to Proposition 1.2.6.1.

By [121, §3, Thm. 2] (which is stated in the simple case but holds in the isotypic

case by the same proof), there exists a number field K ⊂ Qp, a g-dimensional

abelian variety B over K with good reduction at the induced p-adic place v of K, an

embedding of finite fields κ → κv, and an action of OL on B such that the reduction

B0 at v has qv-Frobenius in OL given by the action of πv = π[κv:κ] ∈ OL. (Here,

qv = #κv.) Since B0 admits a CM structure over κv by a field (namely, L), it is κv-

isotypic. Thus, since dim(B0) = dim(B) and Aκv satisfies FrAκv

,qv

=

FrA,q:κ] [κv

= πv,

it follows from the results (i) and (ii) in [118] recalled above that there exists a

κv-isogeny φ : B0 → Aκv .

1.6.5.2. Remark. The κv-isogeny φ : B0 → Aκv at the end of the proof of The-

orem 1.6.5.1 might not be L-linear, though it is Q(πv)-linear since it is compatible

with qv-Frobenius endomorphisms. We can exploit the Q(πv)-linearity of φ to find

an L-linear κv-isogeny B0 → Aκv as follows.