TORSION OF ABELIAN VARIETIES, WEIL CLASSES AND CYCLOTOMIC EXTENSIONS

YURI G. ZARHIN

to appear in Math. Proc. Cambridge Philosoph. Soc.

Let K ⊂ C be a field finitely generated over Q, K(a) ⊂ C the algebraic closure of K, G(K) = Gal(K(a)/K) its Galois group. For each positive integer m we write K(μ_{m}) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime ℓ we write K(ℓ) for the subfield of K(a) obtained by adjoining to K all ℓ−power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab) ⊂ K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K = Q then Q(ab) = Q(c) (the Kronecker-Weber theorem). We write χ_{ℓ} : G(K) → Z_{ℓ}^{∗} for the cyclotomic character defining the Galois action on all ℓ-power roots of unity. We write χ̄_{ℓ} = χ_{ℓ} mod ℓ : G(K) → Z_{ℓ}^{∗} → (Z/ℓZ)^{∗} for the cyclotomic character defining the Galois action on the ℓ−th roots of unity. The character χ_{ℓ} identifies Gal(K(ℓ)/K) with a subgroup of Z_{ℓ}^{∗} = Gal(Q(ℓ)/Q). Let μ(Z_{ℓ}) be the finite cyclic group μ(Z_{ℓ}) of all roots of unity in Z_{ℓ}^{∗}. Its order is equal to ℓ − 1 if ℓ is odd and 2 if ℓ = 2. Let Q(ℓ)′ be the subfield of μ(Z_{ℓ})−invariants in Q(ℓ). Clearly, Gal(Q(ℓ)/Q(ℓ)′) = μ(Z_{ℓ}) and Gal(Q(ℓ)′/Q) = Z_{ℓ}^{∗}/μ(Z_{ℓ}) is isomorphic to Z_{ℓ}.

Let g be a positive integer, X a g−dimensional abelian variety over K. We write End_{K}(X) for the ring of all endomorphisms of X defined over K and End^{0}(X) for the finite-dimensional semisimple Q−algebra End_{K}(X) ⊗ Q. Its center F = F_{X} is a field if and only if X is K−isogenous to a power of a K−simple abelian variety. If so, F is either a totally real number field or a CM-field. We write Lie(X) for the tangent space to X at the origin. It is the g−dimensional K−vector space. By functoriality, End^{0}(X) acts faithfully on Lie(X). We write

Tr_{Lie(X)} : End^{0}(X) ↪ End_{K}(Lie(X)) → K ⊂ C

for the corresponding trace map. The embedding End^{0}(X) ↪ End_{K}(Lie(X)) gives rise to a natural structure of (not necessarily faithful) End^{0}(X) ⊗_{Q} K−module on Lie(X).

The well-known Mordell-Weil-Néron-Lang theorem asserts that X(K) is a finitely generated commutative group. In particular, its torsion subgroup TORS(X(K)) is finite. Hereafter we will write TORS(A) for the torsion subgroup of a commutative group A. This implies that TORS(X(L)) is finite for any finite algebraic extension L of K. Mazur [7] has raised the question of whether the groups X(K(ℓ)) are finitely generated. In this connection, Serre (in letters to Mazur, of January 1974) and Imai [5] have proved independently that TORS(X(K(ℓ))) is finite for all ℓ.

Supported by the NSF.

Now assume that L ⊂ K(a) ⊂ C is an infinite Galois extension of K. When L = K(c) a theorem of Ribet [10] asserts that TORS(X(K(c)) is finite. The author [20] has proven that if the center F of End^{0}(X) is a direct sum of totally real number fields and TORS(X(L)) is infinite then L contains infinitely many roots of unity. On the other hand, Bogomolov (Séminaire Delange-Pisot-Poitou, mai 1982, Paris) proved that TORS(X(L)) is finite if the intersection of L and K(ab) has finite degree over K. For example, if K = Q, we obtain that if TORS(X(L)) is infinite then the intersection of L and Q(c) has infinite degree over Q. The main result of the present paper is the following statement, which deals with essentially non-cyclotomic extensions and may be viewed as a partial improvement of the Bogomolov’s result.

Definition 0.1. We say that X and K satisfy hypothesis (H) if they enjoy one of the following properties:

1. There is a discrete valuation v on K such that X has potential purely multiplicative reduction at v;

2. K does not contain a CM-field (e.g., K ⊂ R);

3. The Hodge group of X is semisimple.

4. The center F of End^{0}(X) is a CM-field and the pair (X, F) is of Weil type, i.e., the F ⊗_{Q} K−module Lie(X) is free.

Remark 0.2. It is proven in [14] that if an abelian variety has somewhere a (potential) purely multiplicative reduction then its Hodge group is semisimple.

Theorem 0.3 (Main Theorem). Let X be a g-dimensional abelian variety over K. Assume that X and K satisfy hypothesis (H). If the intersection of L and K(c) has finite degree over K then TORS(X(L)) is finite.

Remark 0.4. If L is totally real then TORS(X(L)) is finite for an arbitrary X [23]. We refer to [10], [18],[20], [17], [21], [22], [23] for other results concerning the torsion in infinite extensions.

The Main Theorem is an immediate corollary of the following statement.

Theorem 0.5. Let g be a positive integer. There exists a positive integer N = N(g) depending only on g and enjoying the following properties:

Let X be a g−dimensional abelian variety over K and assume that X and K satisfy hypothesis (H).

Then:

• Assume that for some prime ℓ the ℓ−primary part of TORS(X(L)) is infinite. Then K(ℓ) has finite degree over the intersection L ⋂ K(ℓ) and this degree divides (N, ℓ − 1) if ℓ is odd and divides 2 if ℓ = 2 respectively. In addition, L contains Q(ℓ)′.

• Let P = P (X, L) be the set of primes ℓ such that X(L) contains a point of order ℓ. If P is infinite then for all but finitely many primes ℓ ∈ P the degree [K(μ_{ℓ}) : L ⋂ K(μ_{ℓ})] of the field extension K(μ_{ℓ})/L ⋂ K(μ_{ℓ}) divides (N, ℓ−1).

We will prove Theorem 0.5 in Section 3.

I would like to thank the MPI für Mathematik for its hospitality.

1. Main construction

Let F be the center of End_{K}(X) ⊗ Q, R_{F} = F ⋂ End_{K}(X) the center of End_{K}(X). We put

V_{Z} = V_{Z}(X) = H_{1}(X(C), Z), V = V (X) = H_{1}(X(C), Q) = V_{Z} ⊗ Q.

For each nonnegative integer m one may naturally identify the mth rational cohomology group H^{m}(X(C), Q) of X(C) with Hom_{Q}(Λ_{Q}^{m}(V (X), Q). For each prime ℓ there are natural identifications

X_{ℓ} = V_{Z}/ℓV_{Z}, T_{ℓ}(X) = V_{Z} ⊗ Z_{ℓ}, V_{ℓ}(X) = V (X) ⊗_{Q} Q_{ℓ} = V_{Z} ⊗ Q_{ℓ}.

There is a natural Galois action

ρ_{ℓ} = ρ_{ℓ,X} : G(K) → Aut_{Z_{ℓ}}(T_{ℓ}(X)) ⊂ Aut_{Q_{ℓ}}(V_{ℓ}(X)),

induced by the Galois action on the torsion points of X [11]. One may naturally identify the mth ℓ−adic cohomology group H^{m}(X_{a}, Q_{ℓ}) of X_{a} = X × K(a) with

Hom_{Q_{ℓ}}(Λ_{Q_{ℓ}}^{m}(V_{ℓ}(X), Q_{ℓ}) = Hom_{Q}(Λ_{Q}^{m}(V (X), Q)) ⊗_{Q} Q_{ℓ}).

This identification is an isomorphism of the Galois modules.

Assume now that F is a number field, i.e., X is either simple or is isogenous over K to a self-product of a simple abelian variety. Let O_{F} be the ring of integers in F. It is well-known that R_{F} is a subgroup of finite index in O_{F}. Recall that for each prime ℓ there is a splitting F ⊗_{Q} Q_{ℓ} = ⊕F_{λ} where λ runs through the set of prime ideals dividing ℓ in O_{F} and F_{λ} is the completion of F with respect to λ−adic topology. There is a natural splitting V_{ℓ}(X) = ⊕V_{λ}(X) where

V_{λ}(X) = F_{λ}V_{ℓ}(X) = V (X) ⊗_{F} F_{λ}.

It is well-known that all V_{λ}(X) are G(K)−invariant F_{λ}−vector spaces of dimension 2dim(X)/[F : Q]. We write ρ_{λ,X} for the corresponding λ−adic representation

ρ_{λ,X} : G(K) → Aut_{F_{λ}}V_{λ}(X)

of G(K) [11],[9]. Similarly, for all but finitely many ℓ

R_{F} /ℓR_{F} = O_{F} /ℓO_{F} = ⊕_{λ|ℓ}O_{F} /λ

is a direct sum of finite fields O_{F} /λ of characteristic ℓ. Also, X_{ℓ} = V_{Z}/ℓV_{Z} is a free R_{F} /ℓR_{F} = O_{F} /ℓO_{F} −module of rank 2dim(X)/[F : Q] and there is a natural splitting

X_{ℓ} = V_{Z}/ℓV_{Z} = ⊕_{λ|ℓ}X_{λ}

where X_{λ} = (O_{F} /λ) · X_{ℓ}. Clearly, each X_{λ} is a G(K)−invariant O_{F} /λ−vector space of dimension 2dim(X)/[F : Q]. We write ρ̄_{λ,X} for the corresponding modular representation

ρ̄_{λ,X} : G(K) → Aut_{O_{F} /λ}X_{λ}

of G(K) [9]. Let d be a positive integer and assume that there exists a non-zero 2d−linear form ψ ∈ Hom_{Q}(⊗_{Q}^{2d}V (X), Q), enjoying the following properties.

1. For all f ∈ F; v_{1}, … v_{2d} ∈ V (X)

ψ(fv_{1}, v_{2}, …, v_{2d}) = ψ(v_{1}, fv_{2}, …, v_{2d}) = ⋯ = ψ(v_{1}, v_{2}, …, fv_{2d}).

2. For any prime ℓ let us extend ψ by Q_{ℓ}−linearity to the non-zero multilinear form ψ_{ℓ} ∈ Hom_{Q_{ℓ}}(⊗_{Q_{ℓ}}^{2d}V_{ℓ}(X), Q_{ℓ}). Then for all σ ∈ G(K); v_{1}, … v_{2d} ∈ V_{ℓ}(X)

ψ_{ℓ}(σ(v_{1}), σ(v_{2}), …, σ(v_{2d})) = χ_{ℓ}^{d}(σ)ψ_{ℓ}(v_{1}, v_{2}, …, v_{2d}).

We call such a form admissible or d−admissible.

Example. Let us assume that F is a totally real number field. If L is an invertible sheaf on X defined over K and algebraically non-equivalent to zero then one may associate to L its first Chern class

c_{1}(L) ∈ H^{2}(X(C), Q) = Hom_{Q}(Λ_{Q}^{2}(V (X), Q).

The well-known properties of Rosati involutions and Weil pairings imply that c_{1}(L) is 1−admissible (see p. 237 of [8], especially the last sentence and Section 2 of [15]).

There exists a unique F − 2d−linear form ψ_{F} ∈ Hom_{F} (⊗_{F}^{2d}V (X), F) such that

ψ = Tr_{F/Q}(ψ_{F}).

Multiplying ψ by a sufficiently divisible positive integer, we may and will assume that the restriction of ψ_{F} to V_{Z} × ⋯ V_{Z} takes on values in R_{F}. Let Im(ψ_{F}) be the additive subgroup of R_{F} generated by values of ψ_{F} on V_{Z} × ⋯ V_{Z} takes on values in R_{F}. Let Im(ψ_{F}) be the additive subgroup of R_{F} generated by values of ψ_{F} on V_{Z} × ⋯ V_{Z}. Clearly, Im(R_{F}) is a subgroup of finite index in R_{F} that is an ideal. Notice that for all but finitely many primes ℓ

O_{F} = R_{F} /ℓR_{F} , Im(ψ_{F}) = R_{F} /ℓR_{F} .

Let us extend ψ_{F} by F_{λ}−linearity to the non-zero multilinear form

ψ_{F,λ} ∈ Hom_{F_{ℓ}}(⊗_{F_{λ}}^{2d}V_{λ}(X), F_{λ}).

Then

ψ_{F,λ}(σ(v_{1}), σ(v_{2}), …, σ(v_{2d})) = χ_{ℓ}^{d}(σ)ψ_{F,λ}(v_{1}, v_{2}, …, v_{2d})

for all σ ∈ G(K); v_{1}, … v_{2d} ∈ V_{λ}(X).

Similarly, for all but finitely many ℓ the form ψ_{F} induces a non-zero multilinear form

ψ_{F}^{(ℓ)} ∈ Hom_{R_{F} /ℓR_{F}} (⊗_{R_{F} /ℓR_{F}}^{2d} X_{ℓ}, R_{F} /ℓR_{F})

enjoying the following properties:

• The subgroup of R_{F} /ℓR_{F} generated by all the values of ψ_{F}^{(ℓ)} coincides with R_{F} /ℓR_{F};

• For all σ ∈ G(K); v_{1}, … v_{2d} ∈ X_{ℓ}

ψ_{F}^{(ℓ)}(σ(v_{1}), σ(v_{2}), …, σ(v_{2d})) = χ̄_{ℓ}^{d}(σ)ψ_{F}^{(ℓ)}(v_{1}, v_{2}, …, v_{2d}).

This implies that for all but finitely many ℓ the restriction of ψ_{F}^{ℓ} to X_{λ} defines a non-zero multilinear form

ψ_{F}^{(λ)} ∈ Hom_{O_{F} /λ}(⊗_{O_{F} /λ}^{2d}X_{ℓ}, O_{F} /λ)

enjoying the following property:

ψ_{F}^{(λ)}(σ(v_{1}), σ(v_{2}), …, σ(v_{2d})) = χ̄_{ℓ}^{d}(σ)ψ_{F}^{(λ)}(v_{1}, v_{2}, …, v_{2d})

for all σ ∈ G(K); v_{1}, … v_{2d} ∈ X_{λ}.

Remark 1.1. Using the Künneth formula for X_{a}^{2d}, one may view ψ_{ℓ} as a Tate cohomology class on X_{a}^{2d}. If ψ is skew-symmetric then ψ_{ℓ} is a Tate cohomology class on X_{a}.

Theorem 1.2. Assume that the center F of End^{0}X is a field and there is a d−admissible form ψ on X. Let ℓ be a prime and assume that the ℓ− torsion in X(L) is infinite. If L^{(ℓ)} is the intersection of L and K(ℓ) then the field extension K(ℓ)/L^{(ℓ)} has finite degree dividing (d, ℓ − 1) if ℓ is odd and dividing 2 if ℓ = 2 respectively. In addition, L contains Q(ℓ)′.

Proof. As explained in ([21], 0.8, 0.11) the assumption that the ℓ−torsion in X(L) is infinite means that there exists a place λ of F, dividing ℓ such that the Galois group G(L) of L acts trivially on V_{λ}(X). Since ψ_{F,λ} is not identically zero, we conclude that

χ_{ℓ}^{d}(σ) = 1 ∀σ ∈ G(L) ⊂ G(K).

We write G′ for the kernel of χ_{ℓ}^{d}. We have G(L) ⊂ G′ ⊂ G(K).

Recall that the kernel of χ_{ℓ} : G(K) → Z_{ℓ}^{∗} coincides with the Galois group G(K(ℓ)) of K(ℓ) and χ_{ℓ} identifies Gal(K(ℓ)/K) with a subgroup of Z_{ℓ}^{∗} = Gal(Q(ℓ)/Q). Since the torsion subgroup of Z_{ℓ}^{∗} is the cyclic group μ(Z_{ℓ}) of order ℓ − 1 if ℓ is odd and of order 2 if ℓ = 2, G′ coincides with the kernel of (χ_{ℓ})^{d′} with d′ = (d, ℓ − 1) if ℓ is odd and d′ = (d, 2) if ℓ = 2 respectively. This implies that the field K′ = K(a)^{G′} of G′−invariants is a subfield of K(ℓ) and [K(ℓ) : K′] divides d′, since χ_{ℓ} establishes an isomorphism between Gal(K(ℓ)/K′) and

{s ∈ Im(χ_{ℓ}) ⊂ Z_{ℓ}^{∗} | s^{d′} = 1} ⊂ {s ∈ μ(Z_{ℓ}) | s^{d′} = 1}.

Now it is clear that K′ ⊂ L, since G(L) ⊂ G′ = G(K′). It is also clear that K(ℓ)/K′ is a cyclic extension of degree dividing d′.

In order to prove the last assertion of the theorem, notice that Gal(K(ℓ)/K) ⊂ Gal(Q(ℓ)/Q) = Z_{ℓ}^{∗} and the finite subgroup Gal(K(ℓ)/K′) of Gal(K(ℓ)/K) sits in μ(Z_{ℓ}) ⊂ Z_{ℓ}^{∗}. Since μ(Z_{ℓ}) = Gal(Q(ℓ)/Q(ℓ)′), Q(ℓ)′ ⊂ K′. Since K′ ⊂ L, Q(ℓ)′ ⊂ L.

Theorem 1.3. Assume that the center F of End^{0}X is a field and there is a d− admissible form ψ on X. Let S be an infinite set of primes ℓ such that for all but finitely many ℓ ∈ S the ℓ−torsion in X(L) is not zero. Then for all but finitely many ℓ ∈ S the field extension K(μ_{ℓ})/K(μ_{ℓ}) ⋂ L has degree dividing (d, ℓ − 1).

Proof. For all but finitely many ℓ the G(K)−module X_{ℓ} is semisimple and the centralizer of G(K) in End(X_{ℓ}) coincides with End_{K}(X) ⊗ Z/ℓZ. This assertion was proven in [19] for number fields K; the proof is based on results of Faltings [2]. (See [6] for an effective version.) However, the same proof works for arbitrary finitely generated fields K, if one uses results of [3], generalizing the results of [2]. Clearly, for all but finitely many ℓ the center of End_{K}(X) ⊗ Z/ℓZ coincides with R_{F} /ℓR_{F} = O_{F} /ℓO_{F}. Applying Theorem 5f of [20] to G = G(K), G′ = G(L), H = X_{ℓ}, D = End_{K}(X) ⊗ Z/ℓZ, R = F_{F} /ℓR_{F}, we conclude that for all but finitely many ℓ ∈ S there exists λ | ℓ such that G(L) acts trivially on X_{λ}. Using the Galois equivariance of the non-zero form ψ_{F}^{(λ)}, we conclude that for all but finitely many ℓ ∈ S the character χ̄_{ℓ}^{d} kills G(L). We write G′ for the kernel of χ̄_{ℓ}^{d}. We have G(L) ⊂ G′ ⊂ G(K).

Recall that the kernel of χ̄_{ℓ} : G(K) → (Z/ℓZ)^{∗} coincides with G(K(μ_{ℓ})) and (Z/ℓZ)^{∗} is a cyclic group of order ℓ − 1. This implies that the field K′ = K(a)^{G′} of G′−invariants is a subfield of K(μ_{ℓ}) and [K(μ_{ℓ}) : K′] divides (ℓ − 1, d), since χ̄_{ℓ} establishes an isomorphism between Gal(K(μ_{ℓ})/K′) and {s ∈ Im(χ̄_{ℓ}) ⊂ (Z/ℓZ)^{∗} | s^{d} = 1}. One has only to notice that K′ ⊂ L, since G(L) ⊂ G′ = G(K′).

Corollary 1.4. Assume that the torsion subgroup of X(L) is infinite. Then the intersection of L and K(c) has infinite degree over K.

Proof. Indeed, either there is a prime ℓ such that the ℓ−torsion in X(L) is infinite or for infinitely many primes ℓ the ℓ − torsion in X(L) is not zero. Now, one has only to apply the previous two theorems.

2. Weil classes and admissible forms

Suppose A is an abelian variety defined over K, k is a CM-field, ι : k ↪ End_{K}^{0}(A) is an embedding, and C is an algebraically closed field containing K (for instance, C = C or C = Q̄). Let Lie(A) be the tangent space of A at the origin, an K-vector space. If σ is an embedding of k into C, let

n_{σ} = dim_{C}{t ∈ Lie(A) ⊗_{K} C : ι(α)t = σ(α)t for all α ∈ k}.

Write σ̄ for the composition of σ with the involution complex conjugation of k.

Recall that a triple (A, k, ι) is of Weil type if A is an abelian variety over an algebraically closed field C of characteristic zero, k is a CM-field, and ι : k ↪ End^{0}(A) is an embedding, such that n_{σ} = nσ̄ for all embeddings σ of k into C.

It is known [15] that (A, k, ι) is of Weil type if and only if ι makes Lie(A) ⊗_{K} C into a free k ⊗_{Q} C-module (see p. 525 of [10] for the case where k is an imaginary quadratic field). Now, assume that A = X and the image ι(k) contains the center F of End_{K}(X) ⊗ Q (for instance, F = k). Notice that in the case of Weil type the degree [k : Q] divides dim(A). In particular, dim(A) is even.

Our goal is to construct an admissible form, using a triple (A, k, ι) of Weil type. Recall that the degree [k : Q] divides dim(A), put d = dim(X)/[k : Q] and

consider the space of Weil classes ([16], [1], [15])

W_{k,X} = Hom_{k}(Λ_{k}^{2d}V (X), Q(d)) ↪ Hom_{Q}(Λ_{Q}^{2d}V (X), Q(d)) = H^{2d}(X(C), Q)(d).

Clearly, W_{k,X} carries a natural structure of one-dimensional k−vector space. If fix an isomorphism of one-dimensional Q−vector spaces Q ≅ Q(2d) then one may naturally identify Hom_{Q}(Λ_{Q}^{2d}V (X), Q(d)) with Hom_{Q}(Λ_{Q}^{2d}V (X), Q) and W_{k,X} can be described as the space of all 2d−linear skew-symmetric form ψ ∈ Hom_{Q}(Λ_{Q}^{2d}V, Q) with

ψ(fv_{1}, v_{2}, …, v_{2d}) = ψ(v_{1}, fv_{2}, …, v_{2d}) = ⋯ = ψ(v_{1}, v_{2}, …, fv_{2d})

for all f ∈ F; v_{1}, … v_{2d} ∈ V (X).

Since (X, k.ι) is of Weil type, all elements of W_{k} are Hodge classes by Proposition 4.4 of [1]. Therefore, by Theorem 2.11 of [1] they must be also absolute Hodge cycles; cf. [1].

Lemma 2.1. Let μ_{k} be the finite multiplicative group of all roots of unity in k. There is a continuous character χ_{X,k} : G(K) → μ_{k} ⊂ k^{∗}, enjoying the following properties:

For each prime ℓ the subgroup

W_{k} ⊂ W_{k,X} ⊗_{Q} Q_{ℓ} ⊂ H^{2d}(X(C), Q)(d) ⊗_{Q} Q_{ℓ} = H^{2d}(X_{a}, Q_{ℓ})(d)

is G(K)−stable and the action of G(K) on W_{k} is defined via the character

χ_{X,k} : G(K) → μ_{k} ⊂ k^{∗} = Aut_{k}(W_{k,X}).

Proof. Since all elements of k are endomorphisms of X defined over K, it follows easily that W_{k,X} ⊗_{Q}Q_{ℓ} is G(K)−stable and G(K) acts on W_{k,X} ⊗_{Q}Q_{ℓ} via a certain character χ_{X,k,ℓ} : G(K) → [k ⊗_{Q} Q_{ℓ}]^{∗} = Π_{λ|ℓ}k_{λ}^{∗}.

Let us consider the Q−vector subspace

C_{AH}^{d}(X) ⊂ H^{2d}(X(C), Q)(d) ⊂ H^{2d}(X_{a}, Q_{ℓ})(d)

of absolute Hodge classes. Then C_{AH}^{d}(X) is G(K)−stable and the action of G(K) on C_{AH}^{d}(X) does not depend on ℓ and factors through a finite quotient; cf. [1], Prop. 2.9b. Since W_{k,X} consists of Hodge classes and X is an abelian variety, all Weil classes are absolute Hodge classes, i.e, W_{k,X} ⊂ C_{AH}^{d}(X), [1], Th. 2.11. This implies easily that the subgroup Im(χ_{X,k,ℓ}) is finite and contained in k^{∗}, since the intersection of W_{k,X} ⊗_{Q} Q_{ℓ} and C_{AH}^{d}(X) coincides with W_{k,X}. (In fact, W_{k,X} coincides even with the intersection of W_{k,X} ⊗_{Q} Q_{ℓ} and H^{2d}(X(C), Q)(d).) This implies also that χ_{X,k,ℓ} does not depend on the choice of ℓ. So, we may view χ_{X,k,ℓ} as the continuous homomorphism

χ_{X,k} := χ_{X,k,ℓ} : G(K) → μ_{k} ⊂ k^{∗},

which does not depend on the choice of ℓ.

Let r be the order of the finite group Im(χ_{X,k}). Clearly, r divides the order of μ_{k}. Let us put Y = X^{r} and consider the Künneth chunk

H^{2d}(X(C), Q)(d)^{⊗r} ⊂ H^{2dr}(X(C)^{r}, Q)(dr) = H^{2dr}(Y (C), Q)(dr)

of the 2drth rational cohomology group of Y. One may easily check that the tensor power

W_{k,X}^{⊗r} ⊂ H^{2d}(X(C), Q)(d)^{⊗r} ⊂ H^{2dr}(X(C)^{r}, Q)(dr) = H^{2dr}(Y (C), Q)(dr)

coincides with the space W_{k,Y} of Weil classes on Y attached to the “diagonal” embedding

k → End^{0}(X) ⊂ End^{0}(X^{r}) = End^{0}(Y).

Since the centers of End^{0}(X) and End^{0}(X^{r}) coincide, the image of k in End^{0}(Y) does contain the center of End^{0}(Y).

One may easily check that G(K) acts on W_{k,Y} = W_{k,X}^{⊗r} via the character χ_{X,k}^{r}, which is trivial, i.e., W_{k,Y} consists of G(K)−invariants.

Let us fix an isomorphism of one-dimensional Q−vector spaces Q ≅ Q(2dr) and choose a non-zero element

ψ ∈ W_{k,Y} ⊂ H^{2dr}(Y, Q)(dr) = Hom_{Q}(Λ_{Q}^{2dr}(V (Y), Q).

Then a skew-symmetric 2dr−linear form ψ is admissible.

Applying to ψ the theorems of the previous section, we obtain the following statement, which implies the case 4 (in the hypothesis (H)) of Theorem 0.5.

Theorem 2.2. Assume that the center F of End^{0}X is a CM-field and (X, F, id) is of Weil type. Let us put d = #(μ_{F}) × 2dim(X)/[F : Q] ∈ Z_{+}. Let L be an infinite Galois extension of K.

1. Let ℓ be a prime such that the ℓ−torsion in X(L) is infinite. Let L^{(ℓ)} be the intersection of L and K(ℓ). Then the field extension K(ℓ)/L^{(ℓ)} has finite degree dividing (d, ℓ − 1) if ℓ is odd and dividing 2 if ℓ = 2 respectively. In addition, L contains Q(ℓ)′.

2. Let S be the set of primes ℓ such that X(L) contains a point of order ℓ and assume that S is infinite. Then for all but finitely many ℓ ∈ S the field extension K(μ_{ℓ})/K(μ_{ℓ}) ⋂ L has degree dividing (d, ℓ − 1).

Remark 2.3. Since [F : Q] divides 2dim(X) = 2g, one may easily find an explicit positive integer M = M(g), depending only on g and divisible by #(μ_{F}) × 2dim(X)/[F : Q]

3. Proof of Theorem 0.5

We may and will assume that X is K−simple. Then the center F of End^{0}X is either a totally real number field or a CM-field. If F is totally real then the assertion of Theorem 0.5 is proven in [20] with N = 1. So, further we assume that F is a CM-field. We also know that the assertion of Theorem 0.5 is true when (X, F) is of Weil type (Case 4 of Hypothesis (H)).

3.1. Cases 1 and 3 of Hypothesis (H). Enlarging K if necessary, we may and will assume that X is absolutely simple and has semistable reduction. Then, the results of [14] imply that in both cases Hdg_{X} is semisimple. This means that (X, F, id) is of Weil type (cf. for instance [14]). Now, one has only to apply the result of the previous section with d = #(μ_{F}) × 2dim(X)/[F : Q] and get the assertion of Theorem 0.5 with N = M(g).

3.2. Case 2 of Hypothesis (H). We know that the assertion of the theorem is true if (X, F, id) is of Weil type. So, we may assume that (X, F, id) is not of Weil type.

Let us consider the trace map

Tr_{Lie(X)} : F ⊂ End^{0}(X) ↪ End_{K}(Lie(X)) → K ⊂ C.

Our assumption means that the image Tr_{Lie(X)}(F) is not contained in R. On the other hand, let us fix an embedding of F into C and let L be the normal closure of F into C. Clearly, L is a CM-field, containing Tr_{Lie(X)}(F). Since Tr_{Lie(X)}(F) ⊂ K, the intersection L ⋂ K contains an element, which is not totally real. Since any subfield of a CM-field is either totally real or CM, the field L ⋂ K is a CM-subfield of K.

Remark 3.1. If K is a number field not containing a CM-field, one may give another proof, using theory of abelian λ−adic representations [11] instead of Weil/Hodge classes. The crucial point is that in this case the Serre’s tori T_{𝔪} are isomorphic to the multiplicative group G_{m} [11], Sect. 3.4.

Corollary 3.2. Let X be a K−simple abelian variety of odd dimension. Assume that K does not contain a CM-subfield (e.g., K ⊂ R). If X(L) contains infinitely many points of finite order then L contains infinitely many roots of unity.

Proof. In the case of the totally real center F this assertion is proven in ([20], Th.6, p. 142) without restrictions on the dimension. So, in order to prove Corollary, it suffices to check that F is not a CM-field.

Assume that F is a CM-field. Since dim(X) is odd, (X, F, id) is not of Weil type. Now, the arguments, used in the proof of Case 2 imply that K contains a CM-subfield. This gives us a desired contradiction.

Remark 3.3. The assertion of Corollary cannot be extended to the even-dimensional case. In Section 4 we give an explicit counterexample.

Remark 3.4. Let X be a g−dimensional abelian variety that is not necessarily K−simple and let F be the center of End^{0}(X). Assume that

Tr_{Lie(X)}(F) ⊂ R.

Then the assertion of Theorem 0.5 holds true for X. Indeed, if Y is a K−simple abelian subvariety of X and F_{Y} is the center of End^{0}(Y) then one may easily check that either F_{Y} is a totally real number field or (Y, F_{Y} , id) is of Weil type.

4. Example

In this section we construct an abelian surface X over Q and a Galois extension L of Q such that L contains only finitely many roots of unity but X(L) contains infinitely many points of finite order. Of course, the intersection of L and Q(c) is of infinite degree over Q.

4.1. Let E be an elliptic curve over Q without complex multiplication (e. g., j(E) is not an integer). Let us put

Y = {(e_{1}, e_{2}, e_{3}) ∈ E^{3} | e_{1} + e_{2} + e_{3} = 0}.

Clearly, Y is an abelian surface over Q isomorphic to E^{2}. Denote by s an automorphism of Y induced by the cyclic permutation of factors of E^{3}, i.e.,

s(e_{1}, e_{2}, e_{3}) = (e_{3}, e_{1}, e_{2}) ∀ (e_{1}, e_{2}, e_{3}) ∈ Y.

Let C be the cyclic subgroup in Aut(X) of order 3 generated by s.

By a theorem of Serre [12] the homomorphism

ρ_{ℓ,E} : G(Q) → Aut_{Z_{ℓ}}(T_{ℓ}(E)) ≅ GL(2, Z_{ℓ})

is surjective for all but finitely many ℓ. Notice, that the composition

detρ_{ℓ,E} : G(Q) → GL(2, Z_{ℓ}) → Z_{ℓ}^{∗}

coincides with χ_{ℓ} : G(K) → Z_{ℓ}^{∗} [12]. In particular, if Q(E(ℓ^{∞})) is the field of definition of all points on E of ℓ-power order then Q(E(ℓ^{∞}))/Q is the Galois extension with the Galois group GL(2, Z_{ℓ}). In addition, the cyclotomic field Q(ℓ) is the maximal abelian subextension of Q(E(ℓ^{∞})) and the subgroup Gal(Q(E(ℓ^{∞}))/Q(ℓ)) ⊂ Gal(Q(E(ℓ^{∞}))/Q) coincides with SL(2, Z_{ℓ}).

Let us fix such an ℓ, assuming in addition that ℓ − 1 is divisible by 3 but not divisible by 9. Let μ_{3,ℓ} be the group of cubic roots of unity in Z_{ℓ}^{∗}. Then there exists a continuous surjective homomorphism pr_{3} : Z_{ℓ}^{∗} → μ_{3,ℓ}, coinciding with the identity map on μ_{3,ℓ}. These properties determine pr_{3} uniquely.

Let us define field L as a subextension of Q(E(ℓ^{∞})) such that Q(E(ℓ^{∞}))/L is a cubic extension, whose Galois (sub)group coincides with

μ_{3,ℓ} · id = {γ · id | γ ∈ μ_{3,ℓ}} ⊂ GL(2, Z_{ℓ}).

It follows immediately that L is a Galois extension of Q and it does not contain a primitive ℓth root of unity. This implies that 1 and −1 are the only roots of unity in L.

Let us choose a primitive cubic root of unity γ ∈ μ_{3,ℓ} and let ι : μ_{3,ℓ} → C be the isomorphism, which sends γ to s.

Now, let us define X as the twist of Y via the cubic character

κ := ιpr_{3}χ_{ℓ} = ιpr_{3}detρ_{ℓ,E} : G(Q) → μ_{3,ℓ} → C ⊂ Aut(Y).

The Galois module T_{ℓ}(X) is the twist of T_{ℓ}(E)^{2} via κ. Namely,

T_{ℓ}(X) = {(v_{1}, v_{2}, v_{3}) ∈ T_{ℓ}(E) ⊕ T_{ℓ}(E) ⊕ T_{ℓ}(E) | v_{1} + v_{2} + v_{3} = 0}

as the Z_{ℓ}−module but

ρ_{ℓ,X}(σ)(v_{1}, v_{2}, v_{3}) = κ(σ)(ρ_{ℓ,E}(σ)(v_{1}), ρ_{ℓ,X}(σ)(v_{2}), ρ_{ℓ,X}(σ)(v_{3}))

for all σ ∈ G(Q). Now, we construct explicitly G(L)−invariant elements of T_{ℓ}(X). Starting with any v ∈ T_{ℓ}(E), put

w = (γ^{−1}v, γv, v) = (γ^{2}v, γv, v) ∈ T_{ℓ}(E) ⊕ T_{ℓ}(E) ⊕ T_{ℓ}(E).

Clearly, w ∈ T_{ℓ}(X); sw = γw. Let us check that w is G(L)−invariant. Clearly,

G(L) = {σ ∈ G(Q) | ρ_{ℓ,E}(σ) ∈ μ_{3,ℓ} · id}.

Let σ ∈ G(L) with ρ_{ℓ,E}(σ) = ζid, ζ ∈ μ_{3,ℓ}. If ζ = 1 , i.e., ρ_{ℓ,E}(σ) = id then all elements of V_{ℓ}(X) are σ−invariant. Since μ_{3,ℓ} = {1, γ, γ^{−1}}, we may assume that ζ = γ, i.e., ρ_{ℓ,E}(σ) = γ · id and therefore detρ_{ℓ,E}(σ) = γ^{2} = γ^{−1}. Then

ρ_{ℓ,X}(σ)(w) =

ι(pr_{3}(detρ_{ℓ,E}(σ)))(ρ_{ℓ,E}(σ)(γ^{2}v), ρ_{ℓ,E}(σ)(γv), ρ_{ℓ,E}(σ)(v)) = ι(γ^{2})(γw) =

s^{2}(γw) = γs^{2}w = γγ^{2}w = w.

This proves that w is G(L)−invariant.

Now, I claim that X(L) contains infinitely many points, whose order is a power of ℓ. Indeed, starting with a non-divisible element v ∈ T_{ℓ}(E) and identifying the group X_{ℓ^{n}} with the quotient T_{ℓ}(X)/ℓ^{n}T_{ℓ}(X), we get a L−rational point (γ^{2}v, γv, v) mod ℓ^{n}T_{ℓ}(X) ∈ T_{ℓ}(X)/ℓ^{n}T_{ℓ}(X) = X_{ℓ^{n}} of order ℓ^{n}.

5. Another Example

Let K be an imaginary quadratic field with class number 1 and let E be an elliptic curve over Q such that End_{K}(E) = O_{K} is the ring of integers in K. In this section we construct a Galois extension L of K such that E(L) contains infinitely many points of finite order but the intersection of L and K(c) is of finite degree over K (even coincides with K).

We write ι : C → C for the complex conjugation z ↦ z̄. We write R for O_{K}. Clearly, End_{Q}(E) = Z ≠ R. It follows easily that

ι(ux) = ū(ι(x)) ∀x ∈ E(C), u ∈ R.

Notice that K is abelian over Q. Since Q(c) = Q(ab), K ⊂ Q(c) and therefore

K(c) = Q(c).

5.1. Let ℓ be a prime number. We write R_{ℓ} for R ⊗ Z_{ℓ}. It is well-known that T_{ℓ}(E) is a free R ⊗ Z_{ℓ}-module of rank 1 and therefore

End_{R_{ℓ}}(T_{ℓ}(E)) = R_{ℓ}, Aut_{R_{ℓ}}(T_{ℓ}(E)) = R_{ℓ}^{∗}.

Let us consider the corresponding ℓ-adic representation

ρ_{ℓ,E} : G(Q) → Aut_{Z_{ℓ}}(T_{ℓ}(E)) ≅ GL(2, Z_{ℓ}).

Clearly, G_{ℓ} := ρ_{ℓ,E}(G(Q)) is not a subgroup of R_{ℓ}^{∗} = Aut_{R_{ℓ}}(T_{ℓ}(E)) but

H_{ℓ} := ρ_{ℓ,E}(G(K)) ⊆ R_{ℓ}^{∗}.

It is also known ([12], Sect. 4.5) that

H_{ℓ} = R_{ℓ}^{∗}

for all but finitely many primes ℓ. Let us fix such an ℓ, assuming in addition that ℓ is unramified and splits in K. This implies that ℓ = 𝔮𝔮̄ for some 𝔮 ∈ K and

O_{K} = 𝔮 · O_{K} + 𝔮̄ · O_{K}, R_{ℓ} = R_{𝔮} ⊕ R𝔮̄, R_{𝔮} = Z_{ℓ}, R𝔮̄ = Z_{ℓ},
𝔮R_{ℓ} = ℓR_{𝔮} ⊕ R𝔮̄ = ℓZ_{ℓ} ⊕ Z_{ℓ} ⊂ Z_{ℓ} ⊕ Z_{ℓ} = R_{ℓ},
𝔮̄R_{ℓ} = R_{𝔮} ⊕ ℓR𝔮̄ = Z_{ℓ} ⊕ ℓZ_{ℓ} ⊂ Z_{ℓ} ⊕ Z_{ℓ} = R_{ℓ},

R_{ℓ}^{∗} = R_{𝔮}^{∗} × R∗ ; 𝔮̄, R_{𝔮}^{∗} = Z_{ℓ}^{∗}, R∗ ; 𝔮̄ = Z_{ℓ}^{∗}.

We also have

T_{ℓ}(E) = T_{𝔮}(E) ⊕ T𝔮̄(E)

where

T_{𝔮}(E) := R_{𝔮} · T_{ℓ}(E), T𝔮̄(E) := R𝔮̄ · T_{ℓ}(E)

are free Z_{ℓ}-modules of rank 1. This implies that for each positive integer i

𝔮^{i}T_{𝔮}(E) = ℓ^{i}T_{𝔮}(E), 𝔮̄^{i}T_{𝔮}(E) = T_{𝔮}(E),
q̄^{i}T𝔮̄(E) = ℓ^{i}T𝔮̄(E), 𝔮^{i}T𝔮̄(E) = T𝔮̄(E)

and therefore

T_{ℓ}(E)/ℓ^{i}T_{ℓ}(E) = T_{𝔮}(E)/ℓ^{i}T_{𝔮}(E)⊕T𝔮̄(E)/ℓ^{i}T𝔮̄(E) = T_{𝔮}(E)/𝔮^{i}T_{𝔮}(E)⊕T𝔮̄(E)/𝔮̄^{i}T𝔮̄(E).

It is also clear that multiplication by 𝔮^{i} is an automorphism of T𝔮̄(E) and multiplication by 𝔮̄^{i} is an automorphism of T_{𝔮}(E).

It follows easily that a point x ∈ E_{ℓ^{i}} = T_{ℓ}(E)/ℓ^{i}T_{ℓ}(E) satisfies 𝔮^{i}x = 0 (respectively 𝔮̄^{i}x = 0) if and only if x ∈ T_{𝔮}(E)/ℓ^{i}T_{𝔮}(E) = T_{𝔮}(E)/𝔮^{i}T_{𝔮}(E) (respectively x ∈ T𝔮̄(E)/ℓ^{i}T𝔮̄(E) = T𝔮̄(E)/𝔮̄^{i}T𝔮̄(E)).

Let us put

τ := ρ_{ℓ,E}(ι) ∈ G_{ℓ} ⊂ Aut_{Z_{ℓ}}(T_{ℓ}(E)).

Then τ^{2} = id and

τ(R_{𝔮}^{∗} × {1})τ^{−1} = {1} × R∗ ; 𝔮̄ ⊂ R_{ℓ}^{∗}, τ({1} × R∗ ; 𝔮̄)τ^{−1} = R_{𝔮}^{∗} × {1}) ⊂ R_{ℓ}^{∗}.

It is also clear that

τ(T_{𝔮}(E)) = T𝔮̄(E), τ(T𝔮̄(E)) = T_{𝔮}(E).

Let us consider the field K(E(ℓ^{∞})) of definition of all points on E of ℓ-power order. It is the Galois extension of K with the Galois group R_{ℓ}^{∗} = R_{𝔮}^{∗} × R∗ ; 𝔮̄. It is also normal over Q and Gal(K(E(ℓ^{∞}))/Q) = G_{ℓ}, since E is defined over Q and K is normal over Q.

Let us define L as a subextension of K(E(ℓ^{∞}))/K such that

Gal(K(E(ℓ^{∞}))/L) = {1} × R∗ ; 𝔮̄ ⊂ R_{𝔮}^{∗} × R∗ ; 𝔮̄ = R_{ℓ}^{∗} = Gal(K(E(ℓ^{∞}))/K).

One may easily check that L coincides with the field K(E(𝔮^{∞})) of definition of all torsion points on E which are killed by a power of 𝔮. In particular, E(L) contains infinitely many points, whose order is a power of ℓ. Let us consider the field L′ = ι(L). Clearly, K ⊂ L′ ⊂ K(E(ℓ^{∞})) and L′ coincides with the field K(E(𝔮̄^{∞})) of definition of all torsion points on E which are killed by a power of 𝔮̄. It is also clear that

Gal(K(E(ℓ^{∞}))/L) = τ({1}×R∗ ; 𝔮̄)τ^{−1} = R_{𝔮}^{∗} ×{1} ⊂ R_{𝔮}^{∗} ×R∗ ; 𝔮̄ = R_{ℓ}^{∗} = Gal(K(E(ℓ^{∞}).

Since the subgroups {1} × R∗ ; 𝔮̄ and R_{𝔮}^{∗} × {1} generate the whole group R_{ℓ}^{∗} = Gal(K(E(ℓ^{∞}))/K),

L ⋂ ι(L) = L ⋂ L′ = K.

It follows that if M/K is a subextension of L/K such that M is normal over Q then M = K. Since K(c) = Q(c), L ⋂ K(c) = L ⋂ Q(c) is a subfield of Q(c) and therefore is normal (even abelian) over Q. It follows that

L ⋂ K(c) = K.

6. Abelian subextensions

The following statement may be viewed as a variant of Theorem 0.5 for arbitrary abelian varieties over number fields.

Theorem 6.1. Let X be an abelian variety over a number field K. Then:

1. If for some prime ℓ the ℓ−primary part of TORS(X(L)) is infinite then L contains an abelian infinite subextension E ⊂ L such that Gal(E/K) ≅ Z_{ℓ} and E/K is ramified only at divisors of ℓ.

2. Let P = P (X, L) be the set of primes ℓ such that X(L) contains a point of order ℓ. If P is infinite then for all but finitely many primes ℓ ∈ P there exist a finite subextension E^{(ℓ)} ⊂ L such that E^{(ℓ)}/K is a ramified abelian extension which is unramified outside divisors of ℓ. In addition, the degree [E^{(ℓ)} : K] is prime to ℓ and degree [E^{(ℓ)} : K] tends to infinity while ℓ tends to infinity.

Corollary 6.2 (Theorem of Bogomolov). If TORS(X(L)) is infinite then L contains an infinite abelian subextension of K.

Proof of Theorem 6.1. First, we may and will assume that X is K−simple, i.e., the center F of the endomorphism algebra of X is a number field.

Second, there is a positive integer d, enjoying the following property:

If m is a positive integer such that φ(m) ≤ 2g = 2dim(X) then d is divisible by m.

Third, let λ be a prime ideal in O_{F} dividing a prime number ℓ. Then, in the notations of Section 1 the following statement is true.

Lemma 6.3. 1. The composition

π_{λ} := (det_{F_{λ}}ρ_{λ,X})^{d} : G(K) → Aut_{F_{λ}}V_{λ}(X) → F_{λ}^{∗} → F_{λ}^{∗}

is an abelian representation of G(K) unramified outside divisors of ℓ.

2. For all but finitely many λ the composition

π̄_{λ} := (det_{F_{λ}} ρ̄_{λ,X})^{d} : G(K) → Aut_{O_{F} /λ}X_{λ} → (O_{F} /λ)^{∗} → (O_{F} /λ)^{∗}

is an abelian representation of G(K) unramified outside divisors of ℓ.

We will prove Lemma at the end of this section. Now, let us finish the proof of Theorem, assuming validity of Lemma.

First, notice that the ratio

e = 2dim(X)/[F : Q]

is a positive integer. Second, for all but finitely many primes p there exists a finite collection of Weil numbers, i.e., certain algebraic integers {α_{1}, … α_{e}} ⊂ F (a), enjoying the following properties:

• (Weil’s condition) There is a positive integer q > 1 such that q is an integral power of p and all | α_{i} |^{2}= q for all embeddings F (a) ⊂ C.

• For all ℓ ≠ p and λ | ℓ there is a subset S_{λ} ⊂ {1, … e} such that (∏_{i∈S_{λ}} α_{i}) ∈ O_{F} and the group Im(π_{λ}) contains ∏_{i∈S_{λ}} α_{i}.

• For all but finitely many λ the subgroup Im(π̄_{λ}) contains (∏_{i∈S_{λ}} α_{i}) mod λ ∈ (O_{F} /λ)^{∗}.

Indeed, let us choose a prime ideal v in the ring O_{K} of all algebraic integers in K such that X has good reduction at v. Let

F r_{v} ∈ Im(ρ_{λ,X}) ⊂ Aut_{F_{λ}}V_{λ}(X)

be Frobenius element F r_{v} at v (defined up to conjugacy)[11],[9]. Then the set of its eigenvalues belongs to F (a), does not depend on the choice of λ and satisfies all the desired properties with p the residual characteristic of v and q = #(O_{K}/v)([13], Ch. 7, Prop. 7.21 and proof of Prop. 7.23).

Proof of assertion 1. We know that there exists λ dividing ℓ such that the subspace V_{λ}(X) consists of G(L)−invariants. This means that G(L) lies in the kernel of π_{λ}. This implies that the field E′ of ker(π_{λ})−invariants is an abelian subextension of L, unramified outside divisors of ℓ and Gal(E′/K) is isomorphic to Im(π_{λ}). Choosing a collection of Weil numbers attached to prime p ≠ ℓ, we easily conclude that Im(π_{λ}) is an infinite commutative ℓ−adic Lie group [11] and therefore, there is a continuous quotient of Im(π_{λ}), isomorphic to Z_{ℓ}. One has to take as E the subextension of E′ corresponding to this quotient.

Proof of assertion 2. We know that for all but finitely many ℓ ∈ P there exists λ dividing ℓ such that X_{λ} consists of G(L)−invariants. This means that the field E^{(ℓ)} of ker(π̄_{λ})−invariants is an abelian subextension of L, unramified outside divisors of ℓ and Gal(E^{(ℓ)}/K) is isomorphic to Im(π_{λ}). In order to prove that [E^{(ℓ)} : K] tends to infinity, let us assume that there exist an infinite subset P′ ⊂ P and a positive integer D such that #(Gal(E^{(ℓ)}/K)) = [E^{(ℓ)} : K] divides D for all ℓ ∈ P′. This means that

π̄_{λ}^{D} : G(K) → (O_{F} /λ)^{∗}

is a trivial homomorphism for infinitely many λ. In order to get a contradiction, let us choose a collection of Weil numbers {α_{1}, … α_{e}} enjoying the properties described above.

Clearly. for any non-empty subset S ⊂ {1, … e} the product α_{S} := ∏_{i∈S} α_{i} is not a root of unity. In addition, if α_{S} ∈ O_{F} then there only finitely many λ such that α_{S}^{D} − 1 is an element of λ. Since there are only finitely many subsets of {1, … 2g}, for all but finitely many λ the group

Im((π̄_{λ})^{D}) ⊂ (O_{F} /λ)^{∗}

contains an element of type α_{S}^{D} mod λ different from 1. This implies that (π̄_{λ})^{D} is a non-trivial homomorphism for all but finitely many λ. This gives the desired contradiction.

Proof of Lemma. Let v be a prime ideal in the ring O_{K} of all algebraic integers in K. We write I_{v} ⊂ G(K) for the corresponding inertia subgroup defined up to conjugacy. Assume that the residual characteristic of v is different from ℓ. It is known [4] that for any σ ∈ I_{v} there exists a positive integer m such that ρ_{ℓ,X}(σ)^{m} is an unipotent operator in V_{ℓ}(X) and its characteristic polynomial has coefficients in Z. This implies that if m is the smallest integer enjoying this property then the characteristic polynomial is divisible by the mth cyclotomic polynomial. This implies that 2g ≥ φ(m) and therefore m divides d. Since V_{λ}(X) is a Galois-invariant subspace of V_{ℓ}(X) and (for all but finitely many ℓ) X_{λ} is a Galois-invariant subspace of T_{ℓ}(X)/ℓT_{ℓ}(X), a Galois automorphism σ^{d} acts as an unipotent operator in V_{λ}(X) and (for all but finitely many λ) in X_{λ}. One has only to recall that the determinant of an unipotent operator is always 1.

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Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA,

Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, 142292, Russia

   E-mail address: zarhin@math.psu.edu