Linear dependence in Mordell-Weil groups

Gajda, Wojciech; Górnisiewicz, Krzysztof

doi:10.1515/crelle.2009.039

Cited by 14 publications

(26 citation statements)

References 6 publications

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“…This is guaranteed by the condition (1) and the first part of condition (3), since F (ζ 8 ) ⊆ F (A [8]). To guarantee that the image of ρ does not lie in one of the subgroups of index 3, it is necessary and sufficient that 3|[F (A[2]) : F ], which is guaranteed by condition (2).…”

Section: Elliptic Curvesmentioning

confidence: 87%

“…In Theorem 3.8, we give a method for computing this density, and carry out this computation when A is a one-dimensional torus (Proposition 4.5) or an elliptic curve (Theorem 5.5 in the non-CM case, and Theorem 5.10 in the CM case). For instance, when A = E is an elliptic curve with complex multiplication, in general α mod p has odd order for a set of p of density 2/9 when 2 splits in the CM ring of E, and 8/15 when 2 is inert in the CM ring of E. That the image of ω(Frob p ) encodes ℓ-power divisibility properties of |α mod p| has already been established for abelian varieties in [27] (see also [8,Proposition 2.11]), where it is shown that various phenomena occur for all primes in a set of positive Dirichlet density. However, no densities are computed for specific varieties.…”

Section: Introductionmentioning

confidence: 94%

“…If ℓ = 2 and β 1 ∈ 2A(T n ) for some n, then Lemma 3.5 implies that F (β 1 ) ⊆ F (A [8]). This contradicts the hypothesis.…”

Section: Torimentioning

confidence: 99%

“…We may thus index elements of GSp(V g,n ) by decompositions (8) and suitable choices for x| Ex and x| Wx , so that Proof. By Lemma A.3, it suffices to prove the result for n = 1.…”

Section: Higher-dimensional Abelian Varietiesmentioning

confidence: 99%

“…In the latter three theorems, for all ℓ = 2 the condition is just α ∈ ℓA(F ). In the cases we consider, this makes explicit [2, Theorem 2, p. 40], which states that if A is an abelian variety or the product of an abelian variety by a torus, then the Kummer part is the full Tate module for all but finitely many ℓ and has open image for all ℓ (see also [27,Theorem 2.8] and [8,Proposition 2.10] for the latter statement). This result stems essentially from work of Ribet [29], [14], where it is shown that for A belonging to a large class of commutative algebraic groups, the modulo-ℓ Kummer part is all of A[ℓ] for all but finitely many ℓ.…”

Section: Introductionmentioning

confidence: 99%

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Galois theory of iterated endomorphisms

Jones

Rouse

2009

Proceedings of the London Mathematical Society

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Abstract. Given an abelian algebraic group A over a global field F , α ∈ A(F ), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F ) with α ∈ 2A(F (A[4])), then the density of p with (α mod p) having odd order is 11/21.

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Section: Elliptic Curvesmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 94%

“…If ℓ = 2 and β 1 ∈ 2A(T n ) for some n, then Lemma 3.5 implies that F (β 1 ) ⊆ F (A [8]). This contradicts the hypothesis.…”

Section: Torimentioning

confidence: 99%

“…We may thus index elements of GSp(V g,n ) by decompositions (8) and suitable choices for x| Ex and x| Wx , so that Proof. By Lemma A.3, it suffices to prove the result for n = 1.…”

Section: Higher-dimensional Abelian Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Galois theory of iterated endomorphisms

Jones

Rouse

2009

Proceedings of the London Mathematical Society

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On a Hasse principle for Mordell–Weil groups

Banaszak

2009

Comptes Rendus. Mathématique

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In this Note we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the Mordell-Weil group of an abelian variety over a number field. To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 347 (2009). Résumé Un principe de Hasse pour les groupes de Mordell-Weil. Dans cette Note, on démontre un principe de Hasse concernant la dépendance linéaire sur Z des points d'ordre infini dans le groupe de Mordell-Weil d'une variété abélienne définie sur un corps de nombres. Pour citer cet article : G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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A counterexample to the local–global principle of linear dependence for Abelian varieties

Jossen

Perucca

2009

Comptes Rendus. Mathématique

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Abstract. Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Gajda in 2002 asked whether it is true that the point P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all but finitely many primes p of k. We answer negatively to Gajda's question.Let A be an abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k). Suppose that for all but finitely many primes p of k the point (P mod p) belongs to (X mod p). Is it true that P belongs to X? This question, which was formulated by Gajda in 2002, was named the problem of detecting linear dependence. The problem was addressed in several papers (see the bibliography) but the question was still open. In this note we show that the answer to Gajda's question is negative by providing a counterexample.Let k be a number field and let E be an elliptic curve without complex multiplication over k such that there are points P 1 , P 2 , P 3 in E(k) which are Z-linearly independent. Let P ∈ E 3 (k) and X ⊆ E 3 (k) be the following:So the group X consists of the images of the point P via the endomorphisms of E 3 with trace zero. Since the points P i are Z-independent and since E has no complex multiplication, the point P does not belong to X. Notice that no non-zero multiple of P belongs to X.Claim. Let p be a prime of k where E has good reduction. The image of P under the reduction map modulo p belongs to the image of X.For the rest of this note, we fix a prime p of good reduction for E. We write κ for the residue field of k at p. To ease notation, we now let E denote the reduction of the given elliptic curve modulo p and write P 1 , P 2 , P 3 , P and X for the image of the given points and the given subgroup under the reduction map modulo p. Our aim is to find an integer matrix M of trace zero such that P = M P in E 3 (κ).

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Linear dependence in Mordell-Weil groups

Cited by 14 publications

References 6 publications

Galois theory of iterated endomorphisms

Galois theory of iterated endomorphisms

On a Hasse principle for Mordell–Weil groups

A counterexample to the local–global principle of linear dependence for Abelian varieties

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