1.6. ABELIAN VARIETIES OVER FINITE FIELDS 73

For the quadratic extension κ of κ, the abelian surface Aκ is isotypic (by

1.2.6.1). But its

q2-Frobenius

is πκ

2

= q, which by the settled case (1) over κ is

the Weil

q2-integer

associated to some supersingular elliptic curve over κ . Hence,

Aκ cannot also be κ -simple, so it is isogenous to E × E for an elliptic curve

E over κ that must be in the isogeny class which occurs in case (1) over κ .

Choosing such an isogeny decomposition provides a non-zero homomorphism Aκ →

E over κ . By the universal property of Weil restriction, we thereby get a non-zero

κ-homomorphism A → Resκ

/κ

(E ). Since A is κ-simple, by dimension reasons

this must be an isogeny. The existence of A is guaranteed by the Honda–Tate

classification, so it follows that for E as in case (1) over κ , the abelian surface

Resκ

/κ

(E ) is necessarily κ-simple and lies in the unique isogeny class occurring in

case (2) over κ.

1.6.2.3. Remark. In the terminology of Theorem 1.3.6.2, the three cases in Corol-

lary 1.6.2.2 correspond to A that are respectively of Type III with e = 1, Type III

with e = 2, and Type IV. We saw in the proof of 1.6.2.2 that the formula for

invv(D) for p-adic places v of Z in case (3) works in cases (1) and (2), and that

the dimension formula in case (3) works in cases (1) and (2) (though these facts in

cases (1) and (2) are also clear by inspection).

The common Q-degree [Z : Q] [D : Z] of maximal commutative subfields of

the division algebra D is 2g in each case of Corollary 1.6.2.2, so simple abelian

varieties over finite fields always have suﬃciently many complex multiplications.

1.6.2.4. Example. Observe that in part (3) of Corollary 1.6.2.2, for any elliptic

curve case that arises necessarily the CM field Z = Q[π] is imaginary quadratic

and D = Z. Writing q = pr, the elliptic curves that arise in this way are as follows,

depending on the behavior of p in the imaginary quadratic field Z = Q[π].

There are several possibilities for the splitting behavior of p in Z: (i) p splits

in Z with π generating the rth power of one of the two primes of Z over p, (ii) p

is inert in Z with r even and π =

pr/2ζ

for an imaginary quadratic root of unity

ζ = ±1 such that p is inert in Q(ζ), or (iii) p is ramified in Z and π generates

the rth power of the unique prime of Z over p. Cases (ii) and (iii) are exactly the

supersingular cases, and since D = Z in theses cases, the geometric endomorphism

algebra is not entirely defined over κ. Hence, part (1) of Corollary 1.6.2.2 gives all

supersingular elliptic curves over κ (up to isogeny) whose geometric endomorphism

algebra is defined over κ.

By passing to products and using Theorem 1.3.4, we obtain the following result.

1.6.2.5. Corollary (Tate). Every abelian variety A over a finite field admits suf-

ficiently many complex multiplications. If A is isotypic then it admits a structure

of CM abelian variety with complex multiplication by a CM field.

1.6.3. Example. As an application of Corollary 1.6.2.2, here are examples of

simple abelian surfaces A over prime fields of any characteristic p ≡ 1 (mod 12)

such that A is not absolutely simple. Let κ be a finite field of size

p2,

with p a prime

such that p ≡ 1 (mod 4) (resp. p ≡ 1 (mod 3)). Choose ζ such that

ζ2

+ 1 = 0

(resp.

ζ2

+ ζ +1 = 0), so Z := Q(ζ) is an imaginary quadratic field of class number