70 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

algebraic de Rham cohomology HdR(A/K)

1

(a filtered K-vector space of dimension

2g). It will be useful in later considerations (e.g., the proof of Theorem 2.2.3) with

lifting problems from positive characteristic to characteristic 0. (Note that when

char(K) = 0, HdR(A/K)

1

provides essentially the same information as the CM type

arising from the L-action on Lie(A) =

H0(A, ΩA/K)∨, 1

in view of the Hodge filtration

on HdR(A/K);

1

cf. Definition 1.5.3.1 and the subsequent discussion there.)

1.6. Abelian varieties over finite fields

In this section we work over a finite field κ with char(κ) = p.

1.6.1. Tate’s theorem and Weil numbers. A fundamental fact in the theory

of abelian varieties over finite fields is:

1.6.1.1. Theorem (Tate’s isogeny theorem). For abelian varieties A and B over

a finite field κ, the natural injective map

Z ⊗Z Hom(A, B) → Hom(A[

∞],B[ ∞])

is bijective for every prime (including = char(κ)).

Proof. By passing to A × B, it suﬃces to treat the case A = B, as we shall now

consider. The case = char(κ) is the main result in [118]; see [82, App. I, Thm. 1]

for a proof as well. Unfortunately, Tate did not publish his proof for the case = p

(though his argument was published in [79]). See Appendix A.1 for a proof.

Tate’s proof of his isogeny theorem is closely tied up with his analysis of the

general structure of endomorphism algebras of abelian varieties over finite fields.

The essential case, and the one on which we will now focus, is a simple abelian

variety A over a finite field κ. In this case D :=

End0(A)

is a division algebra of

finite dimension over Q. If q = #κ then the q-Frobenius endomorphism

π = πA : A −→ A

is central in D since the q-Frobenius is functorial on the category of κ-schemes.

Hence, the number field Q[π] = Q(π) is contained in the center of D.

Even without simplicity or isotypicity hypotheses on A, Tate proved (see [82,

App. I, Thm. 3(a)]) that the commutative Q-algebra Q[π] is the center of

End0(A)

for any abelian variety A over κ.

1.6.1.2. Definition. Let q =

pn

for a positive integer n and prime number p. Let

F be a field of characteristic 0.

(i) A Weil q-integer in F (or a Weil q-integer of weight 1, to be precise) is an

algebraic integer z ∈ F whose Q-conjugates in C have absolute value

q1/2.2

(ii) Let w be an integer. A Weil q-number of weight w is an algebraic number z

such that ordv(z) = 0 for all finite places v of Q(z) prime to q and |τ(y)| =

qw/2 for all injective ring homomorphisms τ : Q(z) → C.

Note that a Weil q-integer as defined above is precisely a Weil q-number of

weight 1 such that ordv(z) 0 for all p-adic places v of Q(z). The interest in

Definition 1.6.1.2 is that Weil proved (see [95, §3]) that for any non-zero abelian

2What

is called a Weil q-integer here is often called a “Weil q-number” or “Weil q-number of

weight 1” in the literature.