1.6. ABELIAN VARIETIES OVER FINITE FIELDS 73
For the quadratic extension κ of κ, the abelian surface 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,
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
E over κ . By the universal property of Weil restriction, we thereby get a non-zero
κ-homomorphism A Resκ

(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

(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
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