1.6. ABELIAN VARIETIES OVER FINITE FIELDS 71

variety A over κ and any = p := char(κ), the Q -linear q-Frobenius action on

V (A) has characteristic polynomial fA,q ∈ Z[T ] that is independent of and has

all roots in C equal to Weil q-integers.

In Tate’s work, he also proved (see [82, App. I, Thm. 3(e)]) that A is isotypic

if and only if the common characteristic polynomial fA,q ∈ Z[T ] of the q-Frobenius

action on the Tate modules has a single monic irreducible factor over Q, in which

case this irreducible factor is obviously the minimal polynomial fπ over Q for the

q-Frobenius endomorphism π ∈

End0(A)

(since π is central). The polynomial fπ

only depends on A through its isogeny class (due to the functoriality of q-Frobenius

on κ-schemes), and by Weil’s Riemann Hypothesis its Gal(Q/Q)-conjugacy class of

roots in C consists of Weil q-integers, where Q is the algebraic closure of Q in C.

1.6.2. The Honda-Tate theorem. Fix an abstract algebraic closure Q of Q

and let Weil(q) denote the set of Weil q-integers in Q. Elements of Weil(q) are

equivalent when they lie in the same Gal(Q/Q)-orbit; i.e., they have the same

minimal polynomial over Q. The following remarkable result relates Weil q-integers

to isogeny classes of simple abelian varieties over a finite field of size q.

1.6.2.1. Theorem (Honda-Tate). Let κ be a finite field of size q. The assignment

A → πA defines a bijection from the set of isogeny classes of simple abelian varieties

over κ to the set of Gal(Q/Q)-conjugacy classes of Weil q-integers.

Proof. We refer the reader to [50], [121], and [95] for a discussion of the proof

of the Honda-Tate theorem. The proof of injectivity in Theorem 1.6.2.1 rests on

the work of Tate related to Theorem 1.6.1.1. The proof of surjectivity uses abelian

varieties in characteristic 0 (in fact, it uses descents to number fields of CM abelian

varieties over C; see Theorem 1.7.2.1). We are not aware of a proof of surjectivity

that avoids abelian varieties in characteristic 0.

The following consequence of the (proof of the) Honda–Tate theorem describes

the possibilities for the division algebra D =

End0(A)

in terms of whether the

center Z is Q or Q(

√

p) (the totally real cases) or a CM field.

1.6.2.2. Corollary. Let A be a simple abelian variety over a finite field κ of size q

and characteristic p. Define D =

End0(A),

so Z := Q(π) is its center. Let π ∈ D

be the q-Frobenius endomorphism. Exactly one of the following occurs.

(1) We have π2 = q = pn with n even. This is precisely the case Z = Q, and occurs

exactly when D is a central quaternion division algebra over Q, in which case

it is the unique quaternion division algebra over Q ramified at exactly {p, ∞}.

Each of the isogeny classes of simple abelian varieties with π ∈ {±pn/2}

consists of supersingular elliptic curves E over κ for which all endomorphisms

of Eκ are defined over κ (equivalently, the geometric endomorphism algebra

End0(Eκ)

coincides with

End0(E)).

(2) We have

π2

= q =

pn

with n odd. This is precisely the case Z = Q(

√

p), and

occurs if and only if D is the unique central quaternion division algebra over

Z ramified at exactly the two infinite places of Z.

The corresponding isogeny class of simple abelian varieties is represented

by the 2-dimensional Weil restriction Resκ /κ(E ) where κ /κ is a quadratic

extension and E is a supersingular elliptic curve over κ whose geometric

endomorphism algebra is defined over κ .