Proof of Tate’s conjecture over finite fields
following Tate and Zarhin
Throughout let k be a finite field and A, B be abelian varieties over k. We also let Γ be
the absolute Galois group of k.
Our purpose is to prove:
Theorem 1 (Tate) Let ℓ be any prime different from the characteristic of k. The canonical
map
Hom (A, B) ⊗ Zℓ −→ HomΓ (Tℓ (A), Tℓ (B))
(1)
is an isomorphism.
1
Facts I’m going to use without proof
• Double centralizer theorem
• Suppose ϕ : A → B is an isogeny and N is included in the kernel. Then ϕ factors
through A/N . Corollary: if ϕ : A → B kills the N -torsion, then there exists ψ : A → B
such that ϕ = [N ] ◦ ψ = ψ ◦ [N ].
• Wedderburn’s theorems on semisimple algebras, and its corollaries:
– ”every right ideal in a semisimple algebra is generated by an idempotent”.
– If k is a field of characteristic zero, k ′ /k is any field extension, and A is a semisimple
k-algebra, then A ⊗ k ′ is a semisimple k ′ -algebra. Corollary: End(A) ⊗ Qℓ is
semisimple
2
Injectivity
We start by showing that (1) is injective.
maps to zero in
P∞ j Take an u ∈ Hom (A, B) ⊗ Zℓ thatP
HomΓ (Tℓ (A), Tℓ (B)). Write u = j=0 ℓ uj , uj ∈ Hom (A, B), and [u]n for nj=0 ℓj uj . Note
that - since Hom (A, B) is a Z-module - [u]n is in Hom (A, B).
Now (since u maps to zero) [u]n is the zero morphism A[ℓn ] → B[ℓn ], so it kills the ℓn torsion. As it is well known, this implies the existence of a certain ψn ∈ Hom (A, B) such that
[u]n = [ℓn ] ◦ ψn . It follows that [u]n belongs to ℓn Hom (A, B) ⊆ ℓn (Hom (A, B) ⊗ Zℓ ), and
since clearly u − un is in ℓn (Hom (A, B) ⊗ Zℓ ) too we find that u lies in ℓn (Hom (A, B) ⊗ Zℓ )
for every n. As Hom (A, B) ⊗ Zℓ is of finite type, this implies that u = 0 as claimed.
1
3
Preliminary reductions
Lemma 2 The cokernel of (1) is torsion-free. In particular, (1) is bijective iff
Hom (A, B) ⊗ Qℓ −→ HomΓ (Tℓ (A), Tℓ (B)) ⊗ Qℓ
(2)
is.
Proof. Let C be the cokernel of (1). If C has torsion elements, then it has elements of exact
order ℓ. Let [c] be such an element and c be a representative of [c] in HomΓ (Tℓ (A), Tℓ (B)).
Now ℓ · c is a morphism ϕ : A → B, and on the other hand it kills the ℓ-torsion of A (since c
sends ℓ-torsion to ℓ-torsion). Therefore there exists a morphism ψ : A → B such that ϕ = ℓψ.
This means ℓ(ψ − c) = 0, and since HomΓ (Tℓ (A), Tℓ (B)) is torsion-free we have ψ = c, i.e.
c = 0, contradicting the fact that c has order ℓ. The rest follows (note that Qℓ is flat over
Zℓ ).
Furthermore, we have the following elementary equivalence (note that the dimension of
the left hand side does not depend on ℓ):
Lemma 3 The following are equivalent:
1. The map (2) is bijective for every ℓ 6= char(k);
2. The map (2) is bijective for one ℓ 6= char(k) and the dimension over Qℓ of the right
hand side HomΓ (Tℓ (A), Tℓ (B)) ⊗ Qℓ does not depend on ℓ.
Set furthermore Vℓ (C) := Tℓ (C) ⊗ Qℓ . Then we have
Lemma 4 The bijectivity of (2) for all A, B is equivalent to that of
Hom (A, A) ⊗ Qℓ −→ HomΓ (Tℓ (A), Tℓ (A)) ⊗ Qℓ ∼
= HomΓ (Vℓ (A), Vℓ (A))
(3)
for all A.
Proof. Take A in this formula to be A × B.
Consider now the two Qℓ -subalgebras of Hom (A, B) ⊗ Qℓ defined by
Eℓ := Image (End(A) ⊗ Qℓ → End Vℓ (A))
and
Fℓ := Image (Γ → End Vℓ (A)) .
Lemma 5 If Fℓ is semi-simple, the bijectivity of (3) is equivalent to the fact that Fℓ is the
commutant of Eℓ in End Vℓ (A).
Proof. Indeed, what we are trying to prove is that Eℓ is the commutant of Fℓ . Now Eℓ is
always semi-simple and Fℓ is semisimple by assumption, so the claim follows from the double
centralizer theorem.
2
4
4.1
Finiteness hypotheses
Strong Version
We are going to denote Hyps (k, A, ℓ) the statement
up to k-isomorphism, there are only finitely many abelian varieties k-isogenous to A via an
isogeny of ℓ-power degree
4.2
Weak Version
We are going to denote Hypw (k, A, ℓ, d) the statement
up to k-isomorphism, there are only finitely many abelian varieties k-isogenous to A via an
isogeny of ℓ-power degree and admitting a k-polarization of degree d2
Note that Strong Finiteness ⇒ Weak Finiteness, and both are implied by the much
stronger statement ”there are only finitely many abelian varieties of a given dimension”.
Tate shows how Weak Finiteness implies his Conjecture. In these notes I’ll try to show
both this, and how Weak Finiteness + Zarhin’s trick together imply Strong Finiteness, which
in turn implies Tate’s conjecture via an easier route.
Remark 6 In my exposé I only discussed the proof that relies on Strong Finiteness.
5
Lemmas on abelian varieties
Lemma 7 Let W be a finite-index, Galois-stable submodule of Tℓ (A). There exists an abelian
variety B and an isogeny f : B → A such that f (Tℓ (B)) = W , where furthermore f has ℓpower degree.
Proof. Choose an n so large that ℓn Tℓ (A) ⊆ W and let N be the image of W in
Tℓ (A)/ℓn Tℓ (A) ∼
= A[ℓn ]. Then N is a Galois-stable submodule of A[ℓn ], and we can set
B := A/N .
We are in the situation of the following diagram:
ℓn
✲ A
✲
A
α
β
✲
A/N
where α exists since N ⊆ ker[ℓn ]. It remains to show that β (Tℓ (B)) = W . From the
above diagram it is clear that Im (α : Tℓ (A/N ) → Tℓ A) ⊇ A[ℓn ], so it is enough to show that
the image of α in Tℓ (A)/ℓn Tℓ (A) = A[ℓn ] is N .
On the other hand, by definition
B[ℓn ] = a ∈ Ators ℓn a ∈ N /N,
and if b ∈ B[ℓn ] is represented by a ∈ A[ℓn ], then α(b) = ℓn a. It is now clear that α maps
onto N .
3
Lemma 8 Let ϕ be an endomorphism of an abelian variety A/k. The characteristic polynomial of ϕ acting on Tℓ (A) has integral coefficients and does not depend on ℓ 6= char(k).
6
Proof using Strong Finiteness
Proposition 9 Suppose Hyps (A, k, ℓ) holds. Then for every Galois-stable submodule W of
Vℓ (A) there exists an u ∈ End (A) ⊗ Qℓ such that u Vℓ (A) = W .
Proof. For every n consider Xn = (Tℓ A ∩ W ) + ℓn Tℓ A. This being open in Tℓ A, we
can find an abelian variety Bn and an isogeny fn : Bn → A such that fn (Tℓ Bn ) = Xn . By
Strong Finiteness, the Bn ’s fall into finitely many isomorphism classes. Therefore we can find
a certain B = Bn0 such that, for infinitely many values of n, there exists an isomorphism
ψn
B −→ Bn . Let I = n ∈ N Bn ∼
= B = Bn0 and consider the diagrams
Bn 0
fn 0
ψn ✲
Bn
❄
✲ Tℓ B n
fn 0 ∼
=
fn
A
ψn
∼
=
Tℓ B n 0
∼
= fn
❄
❄
❄
X n 0 ⊆ Tℓ A
A
X n ⊆ Tℓ A
The element un = fn ◦ ψn ◦ fn−1
exists (or rather, makes sense) in End(A) ⊗ Q, so it
0
also makes sense in End(A) ⊗ Qℓ . On the other hand, every un sends Xn0 to Xn ⊂ Xn0 ,
so un belongs to End (Xn0 ), which is a compact set. So the sequence un has a subsequence
converging to a certain u ∈ End(Xn0 ) ⊂ End Vℓ A. But even better, every un lies in End (A) ⊗
Qℓ , so the same is true for the limit (we are just working with finite-dimensional Qℓ -subspaces,
which are automatically complete).
T
We would now like to prove that uXn0 = ∞
n=0 Xn = Tℓ A ∩ W . Indeed, for any x ∈ Xn0
we have
∞
\
Xn = Tℓ A ∩ W.
u(x) = n→∞
lim un (x) ∈
n∈I
Conversely take any y ∈
∞
\
n=0
n∈I
Xn . For every n ∈ I we can find an xn ∈ Xn such that
n=0
n∈I
un xn = y. This sequence {xn }n admits a converging subsequence xk with limit x ∈ Xn0 . But
now u(x) = limk uk (xk ) = y.
Finally,
[
[
[
u (Vℓ A) =
u (ℓn Xn0 ) =
ℓn · u (Xn0 ) =
ℓn · (Tℓ A ∩ W ) = W.
n∈Z
n∈Z
n∈Z
Theorem 10 If Strong Finiteness holds for A and A2 and a certain ℓ 6= char(k), then:
4
1. Vℓ (A) is a semisimple Qℓ [Γ]-module1 ;
Γ
2. End(A) ⊗ Qℓ ∼
= (End Vℓ (A)) .
Proof. 1. Semisimplicity is equivalent to the existence of an invariant complement. Let W
be a Γ-invariant subspace of Vℓ (A) and
a = u ∈ End(A) ⊗ Qℓ u(Vℓ (A)) ⊆ W .
This is a right ideal in End(A) ⊗ Qℓ . Now End(A) ⊗ Qℓ is a semisimple algebra, so by
Wedderburn’s theorem and its consequences it is generated by an idempotent element e. On
the other hand, aVℓ (A) = W by the previous proposition, so
W = aVℓ (A) = e (End(A) ⊗ Qℓ ) Vℓ (A) = eVℓ (A).
And now we are done: our invariant complement is given by W ′ = (1 − e)Vℓ (A). This is
basic linear algebra (idempotent elements are projectors), so the only thing we need to check
is that W ′ is Γ-invariant. But this is clear: indeed since every γ ∈ Γ commutes with every
e ∈ End (A) ⊗ Qℓ we have, for an element (1 − e)v ∈ W ′ and an automorphism γ ∈ Γ,
γ · (1 − e)v = (1 − e) (γ · v) ∈ (1 − e)Vℓ (A) = W ′ .
2. Let C be the centralizer of End(A) ⊗ Qℓ in End (Vℓ A) and B the centralizer of C. Since
both End (Vℓ A) and End(A) ⊗ Qℓ are semisimple we have B = End(A) ⊗ Qℓ . Furthermore
we have the inclusion End(A) ⊗ Qℓ ⊆ End(Vℓ A)Γ , so it is enough to show that End (Vℓ )Γ ⊆
End(A) ⊗ Qℓ . Take β ∈ End (Vℓ )Γ and let
W = (x, βx) x ∈ Tℓ A
be its graph. As W is a Galois-invariant subspace of Tℓ (A×A) there exists an u ∈ End (A × A)
such that uTℓ (A × A) = W . Also note that End(A
×A) is just M2 (End(A)). Let c be any
c 0
element of C and look at the endomorphism
of A2 . This commutes with u by
0 c
construction, so
c 0
c 0
c 0
(cx, cβx) x ∈ Tℓ A =
W =
uVℓ A = u
Vℓ A ⊂ uVℓ A = W.
0 c
0 c
0 c
However, W is the graph of β, so we are saying that β(cx) = c(βx), i.e. that β commutes
with c, for every c ∈ C. So β ∈ Centralizer(C) = B = End(A) ⊗ Qℓ .
7
Proof using Weak Finiteness
Why is this case worse? Well, basically it turns out we can’t really prove the existence of
projectors directly. Instead we (i.e. Tate) can do this for a restricted class of Galois-invariant
subspaces, and then exploit the double centralizer theorem. In particular, for this section we
set
D := CEnd(Vℓ A) (Eℓ ) .
1
for this we only need Strong Finiteness to hold for A (and ℓ)
5
Lemma 11 Suppose Fℓ is semisimple. Then the bijectivity of (3) is equivalent to D = Fℓ .
Proof. Indeed the bijectivity of (3) says that Eℓ is the centralizer of Fℓ , so the claim follows
from the Double Centralizer theorem and the semisimplicity of Fℓ .
Proposition 12 Asssume Weak Finiteness. For every maximal isotropic (with respect to
a fixed Weil pairing), Galois invariant subspace W of Vℓ A there exists an u ∈ End(A) ⊗ Q
such that uVℓ = W .
Proof. We start as before by defining Xn = (W ∩ Tℓ A) + ℓn Tℓ A, Bn , and fn : Bn → A.
The construction of fn shows it is of degree ℓng .
It is not hard to check (exercise) that since W is maximal isotropic the image of Xn in
Tℓ A/ℓn Tℓ A has order ℓng .
Now Tate’s brilliant remark is that we can equip Bn with polarizations of a fixed degree.
Indeed, consider a polarization θ on A and the pullback fn∗ θ = fˆn θfn of θ to Bn . Note that
this makes sense:
fn
fc
θ
n
cn
Bn −→ A −→ Â −→ B
What’s the degree of this polarization? Well, it’s deg(fn )2 deg θ, so it’s ℓ2ng deg θ, not
good. Now the idea is to prove that we can ”divide” this polarization by ℓn . More precisely,
the Weil pairing induced by this polarization reads
hx, yifn∗ θ = hx, fn∗ θyi = hx, fc
n θfn yi = hfn x, θfn yi = hfn x, fn yiθ
∀x, y ∈ Tℓ (Bn ),
and since the image of fn is Xn we see that
hTℓ (Bn ), Tℓ (Bn )ifn∗ θ ⊆ hXn , Xn iθ ⊆ ℓn hTℓ A, Tℓ Aiθ ⊆ ℓn Zℓ (1).
cn such that
Now I claim that this implies the existence of a polarization ψ : Bn → B
n
n
∗
= ψn ◦ [ℓ ]. Indeed this happens if and only if Bn [ℓ ] is in the kernel of fn θ. Suppose the
contrary, and let x ∈ Bn [ℓn ] be such that y := fn∗ θ(x) 6= 0. Then y is a point of exact order
ℓm for a certain 0 < m ≤ n; the non-degeneracy of the Weil pairing implies that there is a
z ∈ Bn [ℓm ] such that hz, yi = ℓm , a contradiction.
But now the degree of ψn is deg (fn∗ θ) / deg[ℓn ] = deg θ. The rest of the proof goes through
unchanged.
fn∗ θ
Corollary 13 Suppose Fℓ is a product of copies of Qℓ . Then for any subspace W of Vℓ A that
is Galois-stable and isotropic with respect to a fixed Weil pairing we have DW ⊆ W .
Proof. By descending induction. For W maximal follows from the previous proposition:
DW = DuVℓ A = uDVℓ A ⊆ uVℓ A = W,
since D commutes with u by construction.
Otherwise let W ⊥ be the Weil-orthogonal complement of W . Since both the Weil pairing
and the polarization are Galois-equivariant the subspace W ⊥ is Galois-stable. Now use the
assumption on Fℓ : every simple Fℓ -module is a line. Write W ⊥ = W ⊕ W ′ and decompose
both of
Lthem into direct sums of simple Fℓ -modules, i.e. lines. More specifically, write
⊥
W′ = m
i=1 Li . What’s m? Since the Weil pairing is nondegenerate we have m = dim W −
6
dim W = 2g − 2 dim W ≥ 2. Let L1 , L2 be two fixed such lines. Then W ⊕ L1 and W ⊕ L2
are both Galois-stable, isotropic subspaces of Vℓ (A), so by the induction hypothesis they are
both D-stable. Thus their intersection is D-stable as well.
Now something so obvious it’s surprising it could ever be useful:
Lemma 14 Suppose d is an endomorphism of a vector space V with the property that every
v ∈ V is an eigenvector for d. Then d is a scalar.
But note that any Fℓ -stable line L ⊆ Vℓ A is in particular an isotropic subspace, so every
f
d ∈ D acts as a scalar on every Fℓ -stable line. On the other hand, if Fℓ ∼
= Qℓ , then it is in
particular a semisimple
Lf algebra.
Write Vℓ (A) =
i=1 Vi , where Vi is the space on which Fℓ acts through the projection
on the i-th factor. Any vector vi ∈ Vi generates an Fℓ -stable line. It follows that any d ∈ D
acts as a scalar on every Vi , and therefore it belongs to Fℓ . This proves D ⊆ Fℓ , and since
the opposite inclusion is obvious we have finally proved D = Fℓ . By the double centralizer
theorem this shows that (3) is a bijection for this ℓ, and the argument of section 8 concludes
the proof of Tate’s conjecture.
8
The rank of the image of (3) is independent of ℓ
Proposition 15 Let α, β be semi-simple endomorphisms of two vector spaces V, W over a
field K. Suppose the characteristic polynomials of α, β factor over K as
Y
Y
fα =
pa p , fβ =
pb p
(where the products are taken over irreducible, distinct, monic polynomials in K[t]). Then
X
dim γ ∈ Hom (V, W ) γα = βγ =
ap bp deg p =: r(fα , fβ )
p
Moreover this number is invariant under extension of the base field.
Proof. Make V, W into K[x]-modules making x act as α (resp. β). The assumption of
semi-simplicity implies that (as K[x]-modules) we have
Y
Y
V ∼
(K[x]/p)ap , W ∼
(K[x]/p)bp .
=
=
p
p
The lemma now follows easily; for the invariance under extension of the base field notice
that the number in question is the number of common roots of fα , fβ in a fixed algebraic
closure.
In our context this proposition gives
dim HomG (Vℓ A, Vℓ B) = r(fπA , fπB )
as soon as we can check that πA , πB (the Frobenius automorphisms of A and B over k) act
semisimply on the respective Tate modules. However the right hand side does not depend
on ℓ, since the characteristic polynomials of πA , πB are independent of ℓ, so this proves the
statement in the title of this section. We are left with proving
7
Lemma 16 πA acts semisimply on Vℓ A
Proof. It’s enough to show that Q(πA ) is a semisimple algebra (indeed semisimplicity is
stable under separable extension of the base field and that what we are claiming is essentially
that Qℓ (πA ) is semisimple). Note that Q(πA ) injects into End(A). Now any k-rational
endomorphism commutes with πA , so Q(πA ) injects into the center of End(A) ⊗ Q, which is
a product of fields, hence reduced. But a commutative Q-algebra is semisimple exactly when
it is reduced, and we are done.
9
There are primes ℓ for which Fℓ is a product of copies of Qℓ
Let F be the subalgebra of End(A) ⊗ Q generated by πA . This is central and commutative,
hence a product of fields. Moreover since F acts on Vℓ via the topological generator of Γ we
have Fℓ = F ⊗ Qℓ . This now becomes a standard exercise in algebraic number theory (or a
consequence of Chebotarev...)
10
Weak Finiteness can be leveraged to Strong Finiteness
Theorem 17 There are only finitely many abelian varieties of dimension g over k. In particular, Strong Finiteness is true.
Proof. The set of 8g-dimensional abelian varieties over k of the form (X × X t )4 is finite,
up to k-isomorphism: indeed these are included in the set of abelian varieties of dimension
8g that admit a principal polarization, which is finite by the argument of section 11. Clearly,
X is isomorphic over k to an abelian subvariety of (X × X t )4 . In order to finish the proof,
one has only to recall that the set of abelian subvarieties of a given abelian variety is finite,
up to a k-isomorphism (see [1]).
11
There are only finitely many abelian varieties of fixed dimension and fixed polarization degree
A very sketchy proof is as follows (cf. [2] Mumford for more details on the relevant facts
on abelian varieties). Note first that a polarization θ of degree d2 ”is” a line bundle, in the
following sense. Let P be the Poincaré bundle on A × A∨ and (1, θ) be the map (1, θ) : A →
A × A∨ . By definition of a polarization, the line bundle
L = (1, θ)∗ P
g d−1
is ample. The line bundle L⊗3 is then known
tobe very ample, and the embedding A ֒→ P3
⊗3
associated to the complete linear system L is of degree g!3g d.
On the other hand,
Theorem 18 Suppose k is perfect. The k-rational g-cycles of degree d in PN are parametrized
by the rational points of a projective k-variety Chow(g, N, k).
The result we’re after then follows from the fact that Chow (g, 3g d − 1, g!3g d) exists and
is of finite type over k.
8
References
[1] H.W. Lenstra, Jr., F. Oort, Yu. G. Zarhin, Abelian subvarieties, J. Algebra 80 (1996),
513–516.
[2] D. Mumford, Abelian varieties, 2nd edition, Oxford University Press, 1974.
9