Detecting linear dependence by reduction maps

Banaszak, Grzegorz; Gajda, Wojciech; Krasoń, Piotr

doi:10.1016/j.jnt.2005.01.008

Cited by 28 publications

(64 citation statements)

References 17 publications

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“…Khare applied this theorem to prove that every family of one-dimensional strictly compatible l-adic representations comes from a Hecke character. Theorem 1.1 strengthens the results of [2,8,19]. Namely T. Weston [19] obtained a result analogous to Theorem 1.1 with coefficients in Z for R commutative.…”

Section: Introductionsupporting

confidence: 73%

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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Section: Introductionsupporting

confidence: 73%

On a Hasse principle for Mordell–Weil groups

Banaszak

2009

Comptes Rendus. Mathématique

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“…Note that the proofs of Lemmas 2.12 and 2.13 in [BGK3] were done for the number field case, but the function field case can be treated analogously.…”

Section: Kummer Theory and Intermediate Jacobiansmentioning

confidence: 99%

“…We apply the method of the proof of Theorem 3.1 of [BGK3]. For the convenience of the reader we divide the proof into five steps.…”

Section: Proof Of the Theoremmentioning

confidence: 99%

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On reduction map for étale $K$-theory of curves

Banaszak

Gajda

Krasoń

2005

Homology, Homotopy and Applications

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In this paper we investigate reduction of nontorsion elements in theétale K-theory of a curve X over a global field F. We show that the reduction map can be well understood in terms of Galois cohomology of l-adic representations.Dedicated to Victor Snaith on the occasion of his 60-th birthday. IntroductionAssume that X is a smooth, proper and geometrically irreducible curve of genus g, defined over a global field F. Let l be an odd rational prime different from the characteristic of the field F. In the function field case we fix the set of places at infinity. Let n > 0 and let S l be the set of places of F which consists of places of bad reduction of X, places at infinity, and in the number field case, primes lying over l. Denote by X a smooth and proper model of X over the ring of S l -integers of[DF]. For places v ∈S l , we consider the reduction map:induced onétale K-theory by the injection X v → X of the special fiber at v. Theorem . Let J be the Jacobian variety of X and assume that End

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“…This question, which was formulated by Gajda and by Kowalski in 2002, was named the problem of detecting linear dependence. The problem was addressed in several papers [1][2][3][4]6,[9][10][11][12][13] but the question was still open. In a recent note, [7], the first author stated that the answer to this problem is always affirmative, but this is wrong.…”

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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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Detecting linear dependence by reduction maps

Cited by 28 publications

References 17 publications

On a Hasse principle for Mordell–Weil groups

On a Hasse principle for Mordell–Weil groups

On reduction map for étale $K$-theory of curves

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

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