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 = 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.
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