DETECTING LINEAR DEPENDENCE BY REDUCTION MAPS

G. Banaszak, W. Gajda, P. Krasoń

Abstract. We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non-CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-groups of number fields.

1. Introduction.

Let E be an elliptic curve defined over a number field F. By the classical theorems of Mordell and Weil the group E(F) of F -rational points is finitely generated. The torsion part of E(F) can be computed quite efficiently in many cases: for F =ℚ due to results of Lutz, Nagell and Mazur (see, for example [G-SOT]) and for some larger F due to results of Kamienny, Merel and others. One of the basic facts which are useful in calculations of the torsion is the reduction theorem [Si, Prop. 3.1, p.176] which says that the reduction map:

r_{v} : E(F) ⟶ E_{v}(κ_{v})

(where κ_{v}=O_{F} /v) is an injection, when restricted to the torsion subgroup, for all primes v of good reduction. On the other hand, it is well-known that the rank of E(F) is much more difficult to compute (cf. [RS] for an up-to-date survey on the ranks of E(F), when F =ℚ).

One of our goals in this paper is to show that the reduction maps can be used to investigate the nontorsion part of the Mordell-Weil group. Given a finite set of nontorsion points of E(F), one can ask whether it is possible to detect linear dependence among elements of this set by reductions. Similar questions are of interest for the Mordell-Weil group of any higher dimensional abelian variety over F, and also more generally - if we are given a finitely generated abelian group B(F) together with maps r_{v} : B(F) ⟶ B_{v}(κ_{v}) to finite groups - one map for every finite prime of F. A good example of this situation is provided by Quillen’s K-groups K_{2n+1}(F), where n ≥ 0, and the homomorphisms

r_{v} : K_{2n+1}(F) ⟶ K_{2n+1}(κ_{v}).

Typeset by AMS-TEX

induced by projections O_{F} ⟶ κ_{v}. Note that K_{2n+1}(F)=K_{2n+1}(O_{F}), for n > 0 and the K-group is finitely generated cf. [Q1]. Its rank equals the order of vanishing of the Dedekind zeta function of F at s = −n cf. [B]. In this case the target of the reduction map r_{v} is the cyclic group ℤ/(q_{v}^{n+1}−1), where q_{v} denotes the number of elements of the residue field cf. [Q2]. According to the conjecture of Quillen and Lichtenbaum the map to the continuous Galois cohomology

K_{2n+1}(F) ⊗ ℤ_{l} ⟶ H^{1}(G_{F,S_{0}}, ℤ_{l}(n+1))

constructed by Dwyer and Friedlander in [DF], should be an isomorphism, for all odd l. Here we have denoted by G_{F,S_{0}}=G(F_{S_{0}}/F) the Galois group of the maximal extension F_{S_{0}}/F contained in F̄, which is unramified outside of l.

In this paper we apply an axiomatic setup (see Section 2 for the details) which embraces simultaneously the Mordell-Weil groups of elliptic curves (or rather, of all abelian varieties with commutative endomorphism algebras) and K-groups of number fields. Let O be a ring with unity. We use the general framework of the Mordell-Weil systems {B(L)}_{L} of finitely generated left O-modules, indexed by finite extensions L/F, which was developed in [BGK2] for the solution of the support problem for l-adic representations. One part of the structure of the Mordell-Weil system is a collection of well-behaved maps taking values in the Selmer groups defined by Bloch and Kato [BK] in the Galois cohomology of G(F̄/F). In the case of E(F) these maps are induced by the classical Kummer homomorphisms, whereas for K-groups they are given by the Dwyer-Friedlander maps.

Our main results concern linear dependence of nontorsion points in B(F). We assume that the system {B(L)}_{L} and the Galois representation in question meet the assumptions formulated in Section 2.

Theorem A. [Theorem 3.12] Let P and P_{1}, …, P_{r} be nontorsion elements of B(F) such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Denote by Λ the submodule of B(F) generated by P_{1}, P_{2}, …, P_{r}. The following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

In addition to the methods of [BGK2], in the proof of Theorem A we used the Kummer theory over division fields which was developed by Ribet in [Ri]. The main technical result of the paper is Theorem 3.1. In the first part of its proof (Step 1) we applied an argument due to Khare and Prasad (cf. [KP], Lemma 5) extended to the context of the Mordell-Weil systems.

In the case of the K-group K_{2n+1}(F), where n ≥ 0 and F is a number field, we put B(F)=K_{2n+1}(F)/C_{F} , where C_{F} is the subgroup of K_{2n+1}(F) generated by l-parts, of kernels of the Dwyer-Friedlander maps, for all primes l. Note that C_{F} is a finite group by [DF], and if the Quillen-Lichtenbaum conjecture holds true, then B(F)=K_{2n+1}(F). Theorem A applied to this case implies Theorem 4.1, which for two points, i.e., if r=1, was proven already in [BGK1].

Theorem B. [Theorem 4.1] Let P and P_{1}, P_{2}, …, P_{r} be nontorsion elements of K_{2n+1}(F)/C_{F} , such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Let Λ ⊂ K_{2n+1}(F)/C_{F} denote the subgroup generated by P_{1}, P_{2}, …, P_{r}. The following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

If n = 0, then Theorem B gives the statement about the multiplicative group of the field F, which was previously proven by Khare, [K], Proposition 3, p.10.

The problem which we consider in this paper is motivated by the support problem of Erdös and also by our papers [BGK1] and [BGK2] on the generalization of the Erdös problem to l-adic Galois representations. The support problem for an abelian variety A cf. [C-SR, p.277] is the following question: are the points P, P_{1} ∈ A(F) related over the endomorphism algebra of A, if the order of r_{v}(P) ∈ A_{v}(κ_{v}) divides the order of r_{v}(P_{1}), for almost all v ? Note that the groups A_{v}(κ_{v}) are not necessarly cyclic, and therefore our problem for A and r=1, is essentially different from the support problem for A. We would like to mention that Michael Larsen has recently given a solution of the support problem for all abelian varieties cf. [L].

In the case of abelian varieties Theorem A provides the complete solution of the problem of detecting linear dependence of nontorsion points by reductions for the class of abelian varieties for which Serre proved in [Se1] the analog of the open image theorem.

Theorem C. [Theorem 4.2] Let A be a principally polarized abelian variety of dimension g defined over the number field F such that End(A) = ℤ and dim(A) = g is either odd or g = 2 or 6. Let P and P_{1}, …, P_{r} be nontorsion elements of A(F) such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Denote by Λ the subgroup of A(F) generated by P_{1}, P_{2}, …, P_{r}. Then the following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F, where r_{v}: A(F) ⟶ A_{v}(κ_{v}) are the reduction maps.

In particular, the Mordell-Weil group E(F) of any non-CM elliptic curve E defined over the number field F has the property stated in Theorem C, i.e., our initial question has a positive answer for such curves (cf. Corollary 4.3). We hope that in the case of elliptic curves defined over ℚ, Theorem C will find some practical implementations.

While this paper was being prepared we have learned that Tom Weston has proven by a different method a result similar to our Theorem 2.9, for all abelian varieties with commutative algebras cf. [We], Corollary. 2.8.

2. Kummer theory for l-adic representations.

Notation.

l is a prime number F is a number field, O_{F} its ring of integers G_{F} = G(F̄/F)

v denotes a finite prime of O_{F} , κ_{v}=O_{F} /v denotes the residue field at v g_{v} = G(\overline{κ}_{v}/κ_{v}) T_{l} denotes a free ℤ_{l}-module of finite rank d V_{l} = T_{l} ⊗_{ℤ_{l}} ℚ_{l} A_{l} = V_{l}/T_{l} ρ_{l} : G_{F} → GL(T_{l}) is a Galois representation unramified outside a fixed finite

set S_{l} of primes of O_{F} containing all primes above l \overline{ρ_{l^{k}}} denotes the residual representation G_{F} → GL(T_{l}/l^{k}) induced by ρ_{l} F_{l^{k}} = F (A_{l}[l^{k}]), for any k > 0, denotes the number field F̄^{ker\overline{ρ_{l^{k}}}} F_{l^{∞}} = ⋃_{k}F_{l^{k}} G_{l} = G(F_{l}/F) G_{l^{k}} = G(F_{l^{k}}/F) G_{l^{∞}} = G(F_{l^{∞}}/F) H_{l^{k}} = G(\overline{F} /F_{l^{k}}) H_{l^{∞}} = G(\overline{F} /F_{l^{∞}}) C[l^{k}] denotes the subgroup of l^{k}-torsion elements of an abelian group C

C_{l} = ⋃_{k≥1} C[l^{k}], is the l-torsion subgroup of C.

Let L/F be a finite extension and w a finite prime in L. To indicate that w is not over any prime in S_{l} we will write w ∉ S_{l}, slightly abusing notation. Let O be a ring with unity which acts on T_{l} in such a way that the action commutes with the G_{F} -action. Unless specified otherwise (see paragraph after Lemma 3.11), all modules over the ring O, which we consider in this paper, are left O-modules. Let {B(L)}_{L} be a direct system of finitely generated O-modules indexed by all finite field extensions L/F. The structure maps of the system are induced by inclusions of fields. We assume that for every embedding L → L′ of extensions of F, the structure map B(L) → B(L′) is a homomorphism of O-modules. Similarly, for every prime v of F we define a direct system {B_{v}(κ_{w})}_{κ_{w}} of O-modules where κ_{w} is a residue field for a prime w over v in a finite extension L/F. We require that G_{F} acts on both systems: \{B(L)\}_{L} and \{B_{v}(κ_{w})\}_{κ_{w}} functorially. Let us put B(\overline{F}) = \underrightarrow{lim}_{L/F} B(L). We assume that the actions of G_{F} and O have the following properties:

(A_{1}) for each l, each finite extension L/F and any prime w of L, such that w ∉ S_{l}, we have T_{l}^{F r_{w}} = 0, where F r_{w} ∈ g_{w} denotes the arithmetic Frobenius at w.

(A_{2}) for every L and w ∉ S_{l} there are natural maps ψ_{l,L}, ψ_{l,w} and r_{w} respecting G_{F} and O actions such that the diagram commutes:

(2.1)  \begin{matrix}B(L) ⊗ ℤ_{l} & ⟶^{r_{w}} & B_{v}(κ_{w})_{l} \\ ^{ψ_{l,L}}↓ & & ≅↓^{ψ_{l,w}} \\ H_{f,S_{l}}^{1}(G_{L}, T_{l}) & ⟶^{r_{w}} & H^{1}(g_{w}, T_{l}) \end{matrix}

where H_{f,S_{l}}^{1}(G_{L}, T_{l}) is the group defined by Bloch and Kato (cf.[BK], see also [BGK2]). The left (resp., the right) vertical arrow in the diagram (2.1) is an embedding (resp., an isomorphism) for every L (resp., for every w ∉ S_{l})

(A_{3}) for every L the map ψ_{l,L} is an isomorphism for almost all l or B(\overline{F}) is a discrete G_{F} -module divisible by l, for almost all l, and for every L we have: B(\overline{F})^{G_{L}} ≅ B(L) and H^{0}(G_{L}; A_{l}) ⊂ B(L).

For a point R ∈ B(L) (resp., a subgroup Γ ⊂ B(L)) we denote R̂ = ψ_{l,L}(R) (resp. Γ̂=ψ_{l,L}(Γ)). As in [Ri] we impose the following four axioms on the representations which we consider:

(B_{1}) End_{G_{l}}(A_{l}[l]) ≅ O/lO, for almost all l and End_{G_{l^{∞}}} (T_{l}) ≅ O ⊗ ℤ_{l}, for all l

(B_{2}) A_{l}[l] is a semisimple 𝔽_{l}[G_{l}]-module, for almost all l and V_{l} is a semisimple ℚ_{l}[G_{l^{∞}}]-module, for all l

(B_{3}) H^{1}(G_{l}; A_{l}[l]) = 0, for almost all l and H^{i}(G_{l^{∞}}; T_{l}) are finite groups, for all l and all i ≥ 0

(B_{4}) for each finitely generated subgroup Γ ⊂ B(F) the group

Γ′ = {P ∈ B(F) : mP ∈ Γ for some m ∈ ℕ}

is such that Γ′/Γ has a finite exponent.

Note that in the case of the Tate module of an abelian variety, the conditions (A_{1})- (A_{3}) were checked in [BGK2], Section 3. The conditions (B_{1})-(B_{4}) are fulfiled in this case due to theorems of Faltings, Zarhin and Serre (cf. the proof of Theorem 4.2). In the sequel we assume that O is a finitely generated free ℤ-module.

The next lemma describes the relation between conditions in the assumption (B_{1}).

Lemma 2.2. If End_{G_{l}}(A_{l}[l]) ≅ O/lO, then End_{G_{l^{∞}}} (T_{l}) ≅ O ⊗ ℤ_{l}.

Proof. We prove by induction that for every k > 0 we have End_{G_{l^{k}}} (A_{l}[l^{k}]) ≅ O/l^{k}O. Assume that the claim is true for k=1, 2, …, i−1. Consider the commutative diagram:

0 ⟶ End_{G_{li−1}} (A_{l}[l^{i−1}]) ⟶ End_{G_{li}} (A_{l}[l^{i}]) ⟶^{p} End_{G_{l}}(A_{l}[l]) ⟶ 0
≅↑  ^{s}↑  ≅↑
0 ⟶  O/l^{i−1}O  ⟶  O/l^{i}O  ⟶^{p′}  O/lO  ⟶ 0.

The map p in the diagram is a surjection, since p′ is a surjection. Hence, s is an isomorphism. The lemma follows by taking inverse limit over k. □

Let Λ be a finitely generated O-submodule of B(F). Throughout the paper we assume that the points P_{1}, …, P_{r} constitute a basis of Λ over O i.e., they give an O-isomorphism Λ ≅ O^{r}. Let \overline{P}_{i} denote the image of P_{i} in Λ/lΛ. It is clear that \overline{P}_{1}, …, \overline{P}_{r} form a basis of the module Λ/lΛ over O/lO.

In the remainder of this section, following [Ri] we introduce the Kummer theory for the l-adic representations which meet our assumptions. For P ∈ B(F) and k > 0 we have the Kummer maps:

(2.3)  ϕ_{P}^{(k)} : H_{l^{k}} ⟶ A_{l}[l^{k}]

ϕ_{P}^{(k)}(σ) = σ(\frac{1}{l^{k}} P̂) − \frac{1}{l^{k}} P̂.

We define:

Φ^{(k)} : H_{l^{k}} ⟶ ⨁_{i=1}^{r} A_{l}[l^{k}]

Φ^{(k)} = (ϕ_{P_{1}}^{(k)}, ..., ϕ_{P_{r}}^{(k)}).

One checks easily that, for every k > 1, the following diagram commutes:

(2.4)  \begin{matrix}H_{l^{k}} & ⟶^{ϕ_{P}^{(k)}} & A_{l}[l^{k}] \\ ↓ & & ↓^{×l} \\ H_{l^{k−1}} & ⟶^{ϕ_{P}^{(k−1)}} & A_{l}[l^{k−1}]. \end{matrix}

Taking the inverse limits over k in (2.4) we obtain a map:

ϕ_{P}^{(∞)} : H_{l^{∞}} ⟶ T_{l}(A),

which we denote briefly ϕ_{i} := ϕ_{P_{i}}^{(∞)}. We define:

Φ : H_{l^{∞}} ⟶ ⨁_{i=1}^{r} T_{l}

Φ = (ϕ_{1}, …, ϕ_{r}).

Consider the field F_{l}(\frac{1}{l} Λ̂) := \overline{F}^{ker Φ^{(1)}} and the map induced by Φ^{(1)}:

(2.5)  G(F_{l}(\frac{1}{l}Λ̂)/F_{l}) ⟶ ⨁_{i=1}^{r} A_{l}[l]

cf. [BGK2], (4.13). Since \overline{P}_{1}, …, \overline{P}_{r} is a basis of Λ/lΛ over O/lO, the same argument as in [Ri], Theorem 1.2, shows that, for every l, the map (2.5) is an isomorphism of G_{l}-modules.

Lemma 2.6. For almost all l, there exists an isomorphism of O/lO-modules:

(2.7)  Λ/lΛ ⟶ Hom_{G_{l}}(G(F_{l}(\frac{1}{l}Λ̂)/F_{l}); A_{l}[l]).

Proof. By Kummer theory the map P ↦ ϕ_{P} induces a homomorphism:

B(F)/lB(F) ⟶ Hom_{G_{l}}(H_{l}; A_{l}[l]) = H^{1}(H_{l}; A_{l}[l])^{G_{l}}

which fits into the commutative diagram:

(2.8)  \xymatrix{Λ/lΛ \ar@{^{(}->}[d] \ar[r] & Hom_{G_{l}}(G(F_{l}(\frac{1}{l} Λ̂)/F_{l}); A_{l}[l]) \ar@{^{(}->}[d] \\ B(F)/lB(F) \ar[r] \ar@{^{(}->}[d] & Hom_{G_{l}}(H_{l}; A_{l}[l]) \ar[d]_{=} \\ H^{1}(G_{F}; A_{l}[l]) \ar[r]^{res} & H^{1}(H_{l}; A_{l}[l])^{G_{l}}}

as the middle horizontal map. The map (2.7) is the upper horizontal map in the diagram (2.8). The lower horizontal map is the restriction. By the inflation-restriction sequence in Galois cohomology, the kernel of res is contained in H^{1}(G_{l}; A_{l}[l]) which vanishes, for almost all l, by the axiom (B_{3}). The left-lower vertical map in (2.8) is injective by (A_{3}). The left-upper vertical arrow is an injection for almost all l by (B_{4}). This shows that the map (2.7) is an imbedding for almost all l. Comparing the dimensions over ℤ/lℤ, we conclude by the axiom (B_{1}) and the isomorphism (2.5), that the map (2.7) is an isomorphism of O/lO-modules, for almost all l. □

Theorem 2.9. Let P ∈ B(F) be such that for almost all primes v of F, we have r_{v}(P) ∈ r_{v}(Λ). Then there is a natural number a such that aP ∈ Λ.

Proof. Let Λ_{1} be the O-submodule of B(F) generated by P. We consider the Galois

extensions F_{l}(\frac{1}{l} Λ̂)/F and F_{l}(\frac{1}{l} Λ̂_{1})/F, where F_{l}(\frac{1}{l} Λ̂_{1}) := \overline{F}^{ker ϕ_{P}^{(1)}}. Assume that v splits completely in F_{l}(\frac{1}{l} Λ̂)/F. Let w′ be any prime over v in F_{l}(\frac{1}{l} Λ̂)/F. We have κ_{w′} = κ_{v}, hence g_{w′} = g_{v}. Therefore r_{w′}(\frac{1}{l} Λ̂) ∈ H^{1}(g_{v}, T_{l}). Let w be any prime of F_{l}(\frac{1}{l} Λ̂_{1}) over v. Hence, by the assumption r_{w}(\frac{1}{l} Λ̂_{1}) ∈ H^{1}(g_{v}, T_{l}). Let R̂ be any point in \frac{1}{l} Λ̂_{1}. Since F r_{w}(R̂) = R̂ + P̂_{0}, for some P̂_{0} ∈ A_{l}[l], we have:

r_{w}(R̂) = F r_{w}r_{w}(R̂) = r_{w}F r_{w}(R̂) = r_{w}(R̂) + r_{w}(P̂_{0}).

By [BGK2], Lemma 2.13 and the axiom (A_{1}) we obtain P̂_{0} = 0. This means that v splits completely in the field F_{l}(\frac{1}{l} Λ̂_{1}). By the Frobenius density theorem [J], Corollary 5.5, p.136, we get:

(2.10)  F_{l}(\frac{1}{l}Λ̂_{1}) ⊂ F_{l}(\frac{1}{l}Λ̂).

Hence, by the isomorphism (2.7) applied to Λ and Λ_{1}, and the inclusion (2.10) we obtain the diagram:

(2.11)  \begin{matrix}Λ_{1}/lΛ_{1} & ⟶ & Hom_{G_{l}}(G(F_{l}(\frac{1}{l} Λ̂_{1})/F_{l}); A_{l}[l]) \\ ↓ & & ↓^{α} \\ Λ/lΛ & ⟶ & Hom_{G_{l}}(G(F_{l}(\frac{1}{l} Λ̂)/F_{l}); A_{l}[l]), \end{matrix}

where the left vertical arrow is defined to make the diagram (2.11) commute. Since α is injective by (2.10), we obtain an inclusion:

Λ_{1}/lΛ_{1} ⊂ Λ/lΛ.

Put B_{0} = B(F)/Λ. By (2.11) we see that P maps to zero in B_{0}/lB_{0}. Hence, by the axiom (B_{4}) we get that P −Q ∈ (B_{0})_{tor}, for some Q ∈ Λ. This proves the theorem, if we take for a the exponent of the finite group (B_{0})_{tor}. □

Lemma 2.12. If α_{1}, …, α_{r} ∈ O ⊗_{ℤ} ℤ_{l} are such that α_{1}ϕ_{1} + ⋯ + α_{r}ϕ_{r} = 0, then α_{1} = ⋯ = α_{r} = 0.

Proof. Let Ψ be the composition of maps:

B(F) ⊗_{ℤ} ℤ_{l} ↪ H^{1}(G_{F} : T_{l}) ⟶ H^{1}(H_{l^{∞}}; T_{l}).

Note that H^{1}(H_{l^{∞}}; T_{l}) = Hom(H_{l^{∞}}; T_{l}) and Ψ(P_{i} ⊗ 1) = ϕ_{i}. By (B_{3}) we have kerΨ ⊂ (B(F) ⊗_{ℤ} ℤ_{l})_{tor}. Let t := #B(F)_{tor}. Since Ψ is an O⊗_{ℤ}ℤ_{l}-homomorphism, we have:

0 = α_{1}ϕ_{1} + ⋯ + α_{r}ϕ_{r} = Ψ(α_{1}(P_{1} ⊗ 1) + ⋯ + α_{r}(P_{r} ⊗ 1)).

Hence, Σ_{j=1}^{r}α_{j}(P_{j} ⊗ 1) ∈ (B(F) ⊗_{ℤ} ℤ_{l})_{tor}, so:

tα_{1}(P_{1} ⊗ 1) + ⋯ + tα_{r}(P_{r} ⊗ 1) = 0

in B(F) ⊗_{ℤ} ℤ_{l}. Observe that the points P_{1}⊗1, …, P_{r}⊗1 are linearly independent over O ⊗_{ℤ} ℤ_{l} in B(F) ⊗_{ℤ} ℤ_{l}. This implies that tα_{j} = 0 for j = 1, …, r. Hence,

α_{1} = ⋯ = α_{r} = 0,

because O is a free ℤ-module, by assumption. □

Lemma 2.13. The image of the map Φ is open in ⨁_{i=1}^{r} T_{l} in the topology induced from the natural topology on ℤ_{l}.

Proof. It is enough to show that Im Φ has a finite index in ⨁_{i=1}^{r} T_{l}. The proof follows the lines of the proof of [Ri, Th. 1.2]. Let W = ⨁_{i=1}^{r} V_{l} and M = Im(Φ ⊗ 1) ⊂ W, where V_{l}=T_{l}⊗_{ℤ_{l}}ℚ_{l}. Both M and W are ℚ_{l}[G_{l^{∞}}]-modules. First we will show that Φ ⊗ 1 is onto. Suppose it is not. By (B_{2}) we have a nontrivial decomposition W = M ⊕ M_{1} of ℚ_{l}[G_{l^{∞}}]-modules. Let π_{i} : W → V_{l} be a projection that maps M_{1} nontrivially. By the axiom (B_{1}) we have:

π_{i}(v_{1}, …, v_{r}) = ∑_{j=1}^{r} β_{j}v_{j},

for some β_{j} ∈ O ⊗ ℚ_{l}. Since π_{i} is nontrivial, we see that some β_{j} is nonzero. On the other hand

π_{i}(Φ ⊗ 1)(h) = ∑_{j=1}^{r} β_{j}(ϕ_{j}(h) ⊗ 1) = 0,

for all h ∈ H_{l^{∞}}. Since β_{j} ∈ O ⊗ ℚ_{l}, we can multiply the last equality by a suitable power l^{k} to get:

0 = ∑_{j=1}^{r} α_{j}(ϕ_{j}(h) ⊗ 1),

where α_{j} = l^{k}β_{j} ∈ O ⊗ ℤ_{l}. Since the maps:

O ⊗ ℤ_{l} ↪ O ⊗ ℚ_{l}
Hom(G_{l^{∞}}, T_{l}) ↪ Hom(G_{l^{∞}}, V_{l})

are imbeddings of O ⊗ℤ_{l}-modules, we obtain: ∑_{j=1}^{r} α_{j}ϕ_{j} = 0. By Lemma 2.12 this shows:

α_{1} = ⋯ = α_{r} = 0,

a contradiction. Therefore we have M_{1} = 0. Since both ⨁_{i=1}^{r} T_{l} and Im Φ are ℤ_{l}- lattices in ⨁_{i=1}^{r} V_{l}, we see that Im Φ has a finite index in ⨁_{i=1}^{r} T_{l}, as required. □

3. Main Results.

In this section we show that for some {B(L)}_{L}, one can choose a=1 in Theorem 2.9. Let G_{l}^{alg} be the Zariski closure of the image of ρ_{l} in the algebraic group GL_{d}/ℤ_{l}, endowed with the unique structure of the reduced, closed group subscheme. The next proposition is the main technical result of the paper.

Theorem 3.1. Assume that ρ_{l}(G_{F}) is open in G_{l}^{alg}(ℤ_{l}) in the topology induced from ℤ_{l}. In addition, assume that ρ_{l}(G_{F}) contains an open subgroup of the group of homotheties and that the reduction map G_{l}^{alg}(ℤ_{l}) → G_{l}^{alg}(ℤ/l^{k}) is onto, for every k > 0. Let P_{1}, …, P_{r} ∈ B(F) be points of infinite order, which are linearly independent over O. Let

I = {i_{1}, …, i_{s}} ⊂ {1, …, r}

be any subset of indeces and let

J = {j_{1}, …, j_{r−s}} ⊂ {1, …, r}

be such that I ∩ J = ∅ and I ∪ J = {1, …, r}. Then for any natural M, and for any prime l, there are infinitely many primes v, such that the images of the points P_{i_{1}}, …, P_{i_{s}} via the map:

r_{v} : B(F) → B_{v}(κ_{v})_{l}

are trivial and the images of the points P_{j_{1}}, …, P_{j_{r−s}} have orders divisible by l^{M}.

Proof. Step 1. This part of the proof is analogous to the proof of Lemma 5 of [KP]. Let Λ_{I} (resp., Λ_{J}) be the O-submodule of Λ generated by P_{i_{1}}, …, P_{i_{r}} (resp., by P_{j_{1}}, …, P_{j_{r−s}}). Let F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}) (resp., F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{J})) be the composite of the fields F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}) (resp., F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J})), for k > 0. We will use the notation similar to [KP]:

D_{k} = G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})/F_{l^{k}})

E_{k} = G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})/F)

D_{∞} = G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I})/F_{l^{∞}})

E_{∞} = G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I})/F).

Consider the following commutative diagram:

(3.2)  \xymatrix{0 \ar[r] & D_{∞} \ar[r] \ar@{^{(}->}[d] & E_{∞} \ar[r] \ar@{^{(}->}[d] & G_{l^{∞}} \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & (𝔾_{a}^{d})^{r}(ℤ_{l}) \ar[r] & E(ℤ_{l}) \ar[r] & G_{l}^{alg}(ℤ_{l}) \ar[r] & 0.}

The group scheme E = G_{l}^{alg} ⋉ (𝔾_{a}^{d})^{r} is a semi-direct product and (𝔾_{a}^{d})^{r}(ℤ_{l}) ≅ T_{l}^{r} := ⨁_{i=1}^{r} T_{l}. The left vertical arrow in the diagram (3.2) is induced by the map Φ. The image of Φ is open by Lemma 2.13. Since by the assumption G_{l^{∞}} is open in G_{l}^{alg}(ℤ_{l}), it follows by (3.2) that E_{∞} is open in E(ℤ_{l}). Because the map G_{l}^{alg}(ℤ_{l}) → G_{l}^{alg}(ℤ/l^{k}) is onto, the map E(ℤ_{l}) → E(ℤ/l^{k}) is also onto, for k > 0. Consider the following commutative diagram.

(3.3)

[figure: (ℤ/l^{k+1})^{dr}; E(ℤ/l^{k+1}); G_{l}^{alg}(ℤ/l^{k+1}); D_{k+1}; E_{k+1}; G_{l^{k+1}}; (ℤ/l^{k})^{dr}; E(ℤ/l^{k}); G_{l}^{alg}(ℤ/l^{k}); D_{k}; E_{k}; G_{l^{k}}]

Using the argument on the natural congruence subgroup of E(ℤ_{l}) of level l^{k}, as in the proof of Lemma 5, [KP], we note that, for k big enough, the preimage of E_{k} via the middle vertical arrow in the rear wall of diagram (3.3) is E_{k+1}. It follows by the diagram chasing that the left vertical arrow D_{k+1} → D_{k} in the front wall of (3.3) is surjective. This immediately implies that, for k big enough:

(3.4)  F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}) ∩ F_{l^{k+1}} = F_{l^{k}}.

Step 2. We will make use of the following tower of fields. (3.5)

[figure: F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J}); F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J}); F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}); F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}); σ; F_{l^{k+1}}; F_{l^{k}}; id; h; F]

Consider the following commutative diagram:

(3.6)  \xymatrix{0 \ar[r] & G_{1} \ar[r] \ar@{^{(}->}[d] & G_{2} \ar[r] \ar@{^{(}->}[d] & G_{3} \ar[r] \ar@{^{(}->}[d] \ar[r] & 0 \\ 0 \ar[r] & T_{l}^{r−s} \ar[r] & T_{l}^{r} \ar[r] & T_{l}^{s} \ar[r] & 0,}

where we have put:

G_{1} = G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}, \frac{1}{l^{∞}} Λ̂_{J})/F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}))

G_{2} = G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}, \frac{1}{l^{∞}} Λ̂_{J})/F_{l^{∞}})

G_{3} = G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I})/F_{l^{∞}})

and F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}, \frac{1}{l^{∞}} Λ̂_{J}) = F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I})F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{J}). All vertical arrows in the diagram (3.6) are the Kummer maps discussed in Section 2. The right and the middle vertical arrows have open images (equivalently, finite cokernels) by Lemma 2.13. Applying the snake Lemma to the diagram (3.6), we observe that the left vertical arrow has finite cokernel, hence it has an open image. Consider the following commutative diagram:

(3.7)

[figure: G(F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I}, \frac{1}{l^{∞}} Λ̂_{J})/F_{l^{∞}}(\frac{1}{l^{∞}} Λ̂_{I})); T_{l}^{r−s}; G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})); ⨁_{i=1}^{r−s} A_{l}[l^{k}].]

The horizontal arrows in the diagram (3.7) are the Kummer maps. The upper horizontal arrow is the left vertical arrow in the diagram (3.6), so it has an open image. Hence, the lower horizontal arrow in the diagram (3.7) has the cokernel bounded independently of k. Let M be a natural number. It follows that, for k big enough, there is an element σ ∈ G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})), such that σ maps via the horizontal arrow in (3.7) to an element of ⨁_{i=1}^{r−s} A_{l}[l^{k}], whose all r−s projections onto the direct summands A_{l}[l^{k}] have orders divisible by l^{M}.

Step 3. Pick k big enough such that (3.4) holds and there is the element σ, constructed in Step 2. Consider the diagram (3.5). We choose an element

γ ∈ G(F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}) ⊂ G(F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F),

in the following manner: γ|_{F_{l^{k}} ( \frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})} = (σ, Id) in the semidirect product

G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}) = G((F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})) ⋊ G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})/F_{l^{k}})

and γ|_{F_{l^{k+1}}} = h is such that, the action of h on the module T_{l} is given by a nontrivial homothety 1 + l^{k}u_{0}, for some u_{0} ∈ ℤ_{l}^{×}. Such a homothety h exists by the assumption on the image ρ_{l}(G_{F}). By the Chebotarev density theorem there are infinitely many prime ideals v in O_{F} such that γ is equal to the Frobenius element for the prime v in the extension F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F. In the remainder of the proof we work with prime ideals v which we have just selected. For each such v we fix a prime w in F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J}) above v.

Step 4. Let i ∈ I and let l^{c_{i}} be the order of r_{v}(P_{i}) in B_{v}(κ_{v})_{l}, for some c_{i} ≥ 0. Hence, l^{k+c_{i}} \frac{1}{l^{k}} r_{v}(P_{i}) = 0. So Q_{i} = \frac{1}{l^{k}} P_{i} ∈ B(F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I})) maps via the map r_{w_{1}} (see the diagram (2.1)) to the point r_{w_{1}}(Q_{i}) ∈ B_{v}(κ_{w_{1}})_{l} of order l^{k+c_{i}}. Here w_{1} is a prime of F_{l^{k+1}}(\frac{1}{l^{k}} Λ̂_{I}) below w. By the axioms (A_{2}), (A_{3}) and by the choice of v, the point r_{w_{1}}(Q_{i}) comes from an element of B_{v}(κ_{v})_{l}. Using the assumption that the right vertical arrow in the diagram (2.1) is an isomorphism, by the choice of v, we see that the action of h on r_{w_{1}}(Q_{i}) is as follows:

h(r_{w_{1}}(Q_{i})) = (1 + l^{k}u_{0})r_{w_{1}}(Q_{i}).

But r_{w_{1}}(Q_{i}) ∈ B_{v}(κ_{v})_{l}, so h(r_{w_{1}}(Q_{i}))=r_{w_{1}}(Q_{i}), again by the choice of v. Hence, l^{k}r_{w_{1}}(Q_{i})=0. This can only happen, when c_{i} = 0.

Step 5. In this part of the proof we use the argument similar to that in Proposition 2.19 of [BGK2]. Let w_{2} denote the prime in F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J}) below w and let u_{2} denote the prime in F_{l^{k}} below w_{2}. Consider the following commutative diagram:

(3.8)  \begin{matrix}B(F)/l^{k}B(F) & ⟶ & B_{v}(κ_{v})/l^{k}B_{v}(κ_{v}) \\ ↓ & & ↓ \\ B(F_{l^{k}})/l^{k}B(F_{l^{k}}) & ⟶ & B_{v}(κ_{u_{2}})/l^{k}B_{v}(κ_{u_{2}}) \\ ↓ & & ↓ \\ Hom(H_{l^{k}}^{ab}; A_{l}[l^{k}]) & ⟶ & Hom(g_{u_{2}}; A_{l}[l^{k}]). \end{matrix}

The bottom vertical maps are discribed in the natural way (similarly to the map Ψ in the proof of Lemma 2.12). Every point P_{j}, where j∈J, maps via the left vertical arrow in the diagram (3.8) to an element ϕ_{P_{j}}^{(k)} defined by (2.3). The ho-

momorphism ϕ_{P_{j}}^{(k)} factors through the group G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}). We denote

this factorization with the same symbol ϕ_{P_{j}}^{(k)}. By the choice of the element γ in Step 3 (see also the comments following the diagram (3.7)) we see that the element ϕ_{P_{j}}^{(k)}(γ|_{F_{l^{k}} ( \frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})}) ∈ A_{l}[l^{k}] has the order divisible by l^{M}. Hence, the element

ϕ_{P_{j}}^{(k)} ∈ Hom(H_{l^{k}}^{ab}; A_{l}[l^{k}]) has the order divisible by l^{M}, and by the choice of v in Step 3, it maps via the bottom horizontal arrow in the diagram (3.8) to an element of order divisible by l^{M}. So the point P_{j} maps to an element of order divisible by l^{M} via the top horizontal arrow in the diagram (3.8). □

Corollary 3.9. With the same assumptions as in the Theorem 3.1 the Galois group

G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}) ∩ F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J}) / F_{l^{k}})

has the order bounded independently of k.

Proof. In the same way as in Step 2 of the proof of Theorem 3.1 we show that the horizontal Kummer maps in the diagram (3.10) below have cokernels bounded independently of k (see diagram (3.7)).

(3.10)  \xymatrix{G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J})) \ar@{^{(}->}[r] \ar@{^{(}->}[d] & ⨁_{i=1}^{s} A_{l}[l^{k}] \ar@{^{(}->}[d] \\ G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}) \ar@{^{(}->}[r] & ⨁_{i=1}^{r} A_{l}[l^{k}] \\ G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I})) \ar@{^{(}->}[r] \ar@{^{(}->}[u] & ⨁_{i=1}^{r−s} A_{l}[l^{k}] \ar@{^{(}->}[u]}

The diagram (3.10) and the following diagram.

[figure: F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J}); F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}); F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J}); F_{l^{k}}]

show that the subgroup

G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J})) G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}))

of G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}, \frac{1}{l^{k}} Λ̂_{J})/F_{l^{k}}) has a finite index bounded independently of k. Hence, the Galois group

G(F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{I}) ∩ F_{l^{k}}(\frac{1}{l^{k}} Λ̂_{J}) / F_{l^{k}})

has the order bounded independently of k. □

Lemma 3.11. For any finite extension L/F and any prime w ∉ S_{l} in O_{L}, the reduction map

r_{w} : B(L)_{tor} ⟶ B_{v}(κ_{w})

is an imbedding.

Proof. Since (A_{1}) and (A_{2}) are fulfilled by assumption, the lemma follows by the diagram (2.1) and [BGK2], Lemma 2.13. Note that Lemma 2.13 of [BGK2] holds also for l=2. □

Under the assumptions of Section 2, we can derive the following stronger version of Theorem 2.9 in the case when O = ℤ.

Theorem 3.12. Let P and P_{1}, …, P_{r} be nontorsion elements of B(F) such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Denote by Λ the submodule of B(F) generated by the P_{1}, P_{2}, …, P_{r}. The following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

Proof. We prove that (2) implies (1). The opposite implication is obvious. By Theorem 2.9 there exists a ∈ ℤ such that

(3.13)  aP = α_{1}P_{1} + ⋯ + α_{r}P_{r},

for some α_{i} ∈ O. Let l^{M} be the largest power of l that divides a. Fix 1 ≤ i ≤ r. By Theorem 3.1 there are infinitely many primes v such that

r_{v}(P_{1}) = ⋯ = r_{v}(P_{i−1}) = r_{v}(P_{i+1}) = ⋯ = r_{v}(P_{r}) = 0

in B_{v}(κ_{v})_{l} and r_{v}(P_{i}) has order at least l^{M} in B(k_{v})_{l}. Applying r_{v} to the equality (3.13) we get:

(3.14)  ar_{v}(P) = α_{i}r_{v}(P_{i}).

By the assumption of the theorem and the choice of v,

(3.15)  r_{v}(P) = β_{i}r_{v}(P_{i})

in B_{v}(κ_{v})_{l}, for some β_{i} ∈ ℤ. Multipying both sides of (3.15) by a and comparing qith (3.14) we get:

(α_{i}−aβ_{i})r_{v}(P_{i}) = 0

in B_{v}(κ_{v})_{l}, which implies that l^{M} divides α_{i}. So the equality (3.13) gives:

(3.16)  \frac{a}{l^{M}} P = \frac{α_{1}}{l^{M}} P_{1} + ⋯ + \frac{α_{r}}{l^{M}} P_{r} + R,

where R ∈ B(F)[l^{M}]. By Theorem 3.1 we pick infinitely many primes v in F such that r_{v}(P_{1})= … =r_{v}(P_{r})=0 in B_{v}(κ_{v})_{l}, and apply the assumption that r_{v}(P) ∈ r_{v}(Λ), for almost all v. Hence, by applying r_{v} to the equality (3.16), we get r_{v}(R)=0, for infinitely many primes v. This contradicts Lemma 3.11, unless R=0. Hence, we obtain the equality:

\frac{a}{l^{M}} P = \frac{α_{1}}{l^{M}} P_{1} + ⋯ + \frac{α_{r}}{l^{M}} P_{r}.

Repeating the above argument for primes dividing \frac{a}{l^{M}} we finish the proof by induction. □

4. Corollaries of Theorem 3.12.

In the case of K-groups, we put B(F)=K_{2n+1}(F)/C_{F} , for n≥0, where C_{F} is the subgroup of K_{2n+1}(F) generated by l-parts (for all primes l) of kernels of the Dwyer-Friedlander map. Observe that if the Quillen-Lichtenbaum conjecture holds true, then B(F)=K_{2n+1}(F). We obtain the following specialization of Theorem 3.12 which strengthens the Theorem of [BGK1].

Theorem 4.1. Let P and P_{1}, P_{2}, …, P_{r} be nontorsion elements of K_{2n+1}(F)/C_{F} , such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Let Λ ⊂ K_{2n+1}(F)/C_{F} denote the subgroup generated by P_{1}, P_{2}, …, P_{r}. The following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

Proof. Similarly to [BGK2], Example 3.4, let:

ρ_{l} : G_{F} → GL(T_{l}) ≅ ℤ_{l}^{×}

be the one dimensional representation given by the (n+1)th tensor power of the cyclotomic character. For every finite extension L/F, let C_{L} be the subgroup of K_{2n+1}(L) generated by the l-parts (for all odd primes l) of the kernels of the Dwyer-Friedlander maps cf. [DF]:

K_{2n+1}(L) ⟶ K_{2n+1}(L) ⊗ ℤ_{l} ⟶ H^{1}(G_{L}; ℤ_{l}(n + 1)).

Let us put:

B(L) = K_{2n+1}(L)/C_{L}

and c := #C_{F} . If the Quillen-Lichtenbaum conjecture holds true, then c=1. Because T_{l}=ℤ_{l}(n+1) is one-dimensional over ℤ_{l} and B(F) is finitely generated, the axioms (A_{1})−(A_{3}) and (B_{1})−(B_{4}) are clearly satisfied (cf. [BGK2], Section 6). In this case, we have O = ℤ, G_{l}^{alg} = 𝔾_{m} and the map G_{l}^{alg}(ℤ_{l}) → G_{l}^{alg}(ℤ/l^{k}) is the natural projection ℤ_{l}^{×} → (ℤ/l^{k})^{×}. The image of ρ_{l} is open in ℤ_{l}^{×}, for each l, which is clear from the definition of ρ_{l}. □

Theorem 4.2. Let A be a principally polarized abelian variety of dimension g defined over the number field F such that End(A) = ℤ and dim(A) = g is either odd or g = 2 or 6. Let P and P_{1}, …, P_{r} be nontorsion elements of A(F) such that P_{1}, P_{2}, …, P_{r} are linearly independent over ℤ. Denote by Λ the subgroup of A(F) generated by P_{1}, P_{2}, …, P_{r}. Then the following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

Proof. Let ρ_{l} : G_{F} ⟶ GL(T_{l}(A)) be the l-adic representation associated to A. For a finite extension L/F let B(L)=A(L). The axioms (A_{1})−(A_{3}) are satisfied by [BGK2], Examples 3.6, 3.7. The axioms (B_{1}) and (B_{2}) are satisfied by the results of Faltings [F], Satz 4, and Zarhin [Z], Corollary 5.4.5. The condition (B_{3}) holds due to the result of Serre, [Se2], Corollary of Theorem 2, and the condition (B_{4}) holds, because B(F)=A(F) is finitely generated. In this case, we have O=ℤ, G_{l}^{alg}=GSp_{2g} and the map G_{l}^{alg}(ℤ_{l}) → G_{l}^{alg}(ℤ/l^{k}) is the natural map GSp_{2g}(ℤ_{l}) → GSp_{2g}(ℤ/l^{k}), which is surjective, for every k > 0, by [A], Lemma 3.3.2 (1), p.135. The image of ρ_{l} is open in GSp_{2g}(ℤ_{l}), for each l, by [Se1], Théorème 3, p. 97 and it contains an open subgroup of homotheties by the theorem of Bogomolov, cf. [Bo], Corollary 1, p.702. □

Corollary 4.3. Let E be an elliptic curve without complex multiplication, which is defined over the number field F. Let P and P_{1}, …, P_{r} be non-torsion elements of the Mordell-Weil group E(F) and such that P_{1}, …, P_{r} are linearly independent over ℤ. Let Λ be the subgroup generated by P_{1}, …, P_{r}. Then the following two statements are equivalent:

(1) P ∈ Λ

(2) r_{v}(P) ∈ r_{v}(Λ) for almost all primes v of F.

Acknowledgements: The authors thank the Isaac Newton Institute in Cambridge for the financial support during the visit in September 2002. The second author thanks the Max-Planck-Institut für Mathematik in Bonn for the hospitality and the financial support during the visit in 2002/2003. The research was partially financed by a KBN grant 2 PO3A 048 22.

References

[A] A.N. Andrianov, Quadratic forms and Hecke operators, vol. Grundlehren der math. Wissenschaften 286, Springer-Verlag, 1987.

[B] A. Borel, Cohomologie de SL_{n} et values de fonctiones zeta, Ann. Acad. Scuola Normale Superiore 7 (1974), 613-636.

[Bo] F.A. Bogomolov, Sur l’algébricité des représentations l-adiques, vol. 290, C.R.Acad.Sci. Paris Sér. A-B, 1980, pp. A701-A703.

[BGK1] G. Banaszak, W. Gajda, P. Krasoń, A support problem for K-theory of number fields, C. R. Acad. Sci. Paris Sér. 1 Math. 331 no. 3 (2000), 185-190.

[BGK2] G. Banaszak, W. Gajda, P. Krasoń, Support problem for the intermediate jacobians of l-adic representations, to appear in Journal of Number Theory.

[BK] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, I (1990), 333-400.

[C-RS] C. Corralez-Rodrigáñez, R. Schoof, Support problem and its elliptic analogue, Journal

of Number Theory 64 (1997), 276-290.

[DF] W. Dwyer, E. Friedlander, Algebraic and étale K-theory, Trans. Amer. Math. Soc. 292 (1985), 247-280.

[Fa] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zalhkörpern, Inv. Math. 73 (1983), 349-366.

[G-SOT] I. Garcia-Selfa, M.A. Olalla, J.M. Tornero, Computing the rational torsion of an elliptic

curve using Tate normal form, Journal of Number Theory 96 (2002), 76-88.

[J] G. Janusz, Algebraic number theory, Academic Press, London and New York, 1973.

[K] Ch. Khare, Compatible systems of mod p Galois representations, preprint.

[KP] Ch. Khare, D. Prasad, Reduction of abstract homomorphisms of lattices mod p, preprint (2002).

[L] M. Larsen, The support problem for abelian varieties, preprint (2002).

[Q1] D. Quillen, Finite generation of the groups K_{i} of rings of algebraic integers., Lecture Notes in Mathematics 341 (1973), 179-198.

[Q2] D. Quillen, On the cohomology and K-theory of the general linear group over finite field, Ann. of Math. 96 (1972), 552-586.

[Ri] K.A. Ribet, Kummer theory on extensions of abelian varieties by tori, Duke Mathematical Journal 46, No. 4 (1979), 745-761.

[RS] K. Rubin, A. Silverberg, Ranks of elliptic curves, Bull. of the AMS 39 (2002), 455-474.

[Se1] J.-P. Serre, Résumés des cours au Collège de France, Annuaire du Collège de France (1985-1986), 95-99.

[Se2] J.-P. Serre, Sur les groupes de congruence des variétés abéliennes. II, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 731-737.

[Si] J. Silverman, The arithmetic of elliptic curves, GTM 106, Springer-Verlag, 1986.

[We] T. Weston, Kummer theory of abelian varieties and reductions of Mordell-Weil groups, preprint (2002).

[Za] J.G. Zarhin, A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent. math. 79 (1985), 309-321.

Department of Mathematics, Adam Mickiewicz University, Poznań, Poland

   E-mail address: BANASZAK@math.amu.edu.pl

Department of Mathematics, Adam Mickiewicz University, Poznań, Poland

   E-mail address: GAJDA@math.amu.edu.pl

current: Max Planck Intitute für Mathematik in Bonn, Bonn, Germany

   E-mail address: GAJDA@mpim-bonn.mpg.de

Department of Mathematics, Szczecin University, Szczecin, Poland

   E-mail address: KRASON@uoo.univ.szczecin.pl