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 over Q for the
q-Frobenius endomorphism π
End0(A)
(since π is central). The polynomial
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 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 is a quadratic
extension and E is a supersingular elliptic curve over κ whose geometric
endomorphism algebra is defined over κ .
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