Power residues on Abelian varieties

Wong, Siman

doi:10.1007/s002290050008

Cited by 18 publications

(22 citation statements)

References 8 publications

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“…Remark 5.2. Damian Roessler pointed out to the author that Theorem 5.1 can also be deduced from a theorem of Wong [16]. To sketch this set A := A 1 × .…”

Section: Density Results For the Full Reductionmentioning

confidence: 99%

“…Since multiplication by induces an automorphism on the prime-to-part of A v (k v ), the reduction of a i then lies in A v (k v ). Wong [16,Th. 2] deduces from this that at least one a i is contained in A(K).…”

Section: Density Results For the Full Reductionmentioning

confidence: 99%

“…These results as well as Corollary 1.7 are needed in joint work with Damian Roessler [12] and provided the motivation for the present paper. 1 Theorem 5.1 can also be deduced from work by Wong [16] who, instead of studying when the -part of a v is zero, considers the dual question of when a v lies in •A v (k v ). Related questions are addressed in work by Corrales-Rodrigáñez and Schoof [6], Khare and Prasad [10], and Larsen [11].…”

Section: Letmentioning

confidence: 99%

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On the order of the reduction of a point on an abelian variety

Pink

2004

Math. Ann.

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Consider a point of infinite order on an abelian variety over a number field. Then its reduction at any place v of good reduction is a torsion point. For most of this paper we fix a rational prime and study how the-part of this reduction varies with v. Under suitable conditions we prove various statements on this-part for all v in a set of positive Dirichlet density: for example that its order is a fixed power of , that its order is non-trivial for the reductions of finitely many points, or that its order is larger than a certain explicit value that varies with v. By similar methods we prove that for all v in a set of positive Dirichlet density the reduction of a given abelian variety possesses no non-trivial supersingular abelian subvariety.

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“…Remark 5.2. Damian Roessler pointed out to the author that Theorem 5.1 can also be deduced from a theorem of Wong [16]. To sketch this set A := A 1 × .…”

Section: Density Results For the Full Reductionmentioning

confidence: 99%

Section: Density Results For the Full Reductionmentioning

confidence: 99%

Section: Letmentioning

confidence: 99%

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On the order of the reduction of a point on an abelian variety

Pink

2004

Math. Ann.

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“…For these algebraic groups we have an affirmative answer when q is a prime (see [2,Thm 3.1] and [12]) and when q is a power of a prime p / ∈ S = {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163} (see [4,Thm 1]). An interesting open question is if there exists a counterexample for q = p n , with p ∈ S and n > 1.…”

Section: (Gal(k(a[ P])/k) A[ P]) Wherementioning

confidence: 95%

Local–global divisibility by 4 in elliptic curves defined over $${\mathbb{Q}}$$

Paladino

2009

Annali di Matematica

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Let E be an elliptic curve defined over Q. Let P ∈ E(Q) and let q be a positive integer. Assume that for almost all valuations v ∈ Q, there exist pointsIs it possible to conclude that there exists a point D ∈ E(Q) such that P = q D? A full answer to this question is known when q is a power of almost all primes p ∈ N, but some cases remain open when p ∈ S = {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163}. We now give a complete answer in the case when q = 4.

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“…They proved that if p is a prime, an affirmative answer to the problem holds when q = p (see [2], Thm. 3.1,and [20]) and when q = p n , with n ≥ 2 and p / ∈ S = {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163} (see [4], Thm 1). They used a result found by Mazur to count out the primes in S (see [7]).…”

Section: Introductionmentioning

confidence: 99%

Elliptic curves with {\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3) and counterexamples to local-global divisibility by 9

Paladino¹

2010

J. Théor. Nombres Bordeaux

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Laura PALADINOElliptic curves with Q(E[3]) = Q(ζ 3 ) and counterexamples to local-global divisibility by 9 Tome 22, n o 1 (2010), p. 139-160. © Université Bordeaux 1, 2010, tous droits réservés.L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http:// jtnb.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. De plus, nous considérons le problème de la divisibilité localeglobale par 9 pour les points d'une courbe elliptique. Le nombre 9 est une des rares puissances d'un nombre premier pour laquelle on ne connait pas la réponse à la divisibilité locale-globale dans le cas de tels groupes algébriques. Dans ce papier nous donnons une réponse négative. Nous exhibons des courbes de la famille F h,β , avec des points qui sont localement divisibles par 9 presque partout, mais qui ne sont pas globalement divisibles par 9, sur un corps de nombres de degré au plus 2 sur Q(ζ 3 ).Abstract. We give a family F h,β of elliptic curves, depending on two nonzero rational parameters β and h, such that the following statement holds: let E be an elliptic curve and let E[3] be its 3-torsion subgroup. This group verifiesFurthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer to the local-global divisibility is unknown in the case of such algebraic groups. In this paper, we give a negative one. We show some curves of the family F h,β , with points locally divisible by 9 almost everywhere, but not globally, over a number field of degree at most 2 over Q(ζ 3 ). IntroductionLet k be a number field and let A be a commutative algebraic group over k. Let P ∈ A(k). We denote by M k the set of the places v ∈ k and by k v the completion of k at the valuation v. We consider the following question: This problem is known as Local-Global Divisibility Problem. There are known solutions in many cases, but many cases remain open too. By using the Bézout identity , it turns out that it suffices to solve it in the case when q is a power p n of a prime p, to get answers for a general integer q.When A(k) = G m a solution is classical. The answer is affirmative for all odd prime powers q and for q|4 (see [1], Chap IX, Thm. I). On the other hand, there are counterexamples for q = 2 t , t ≥ 3. The most famous of them was discovered by Trost (see [16]) and it is the diophantine equation x 8 = 16, that has a solution in Q p , for all primes p ∈ Q, different from 2, but has no solutions in Q 2 and in Q. This is in accordance with the more general Grunwald-Wang theorem (see [6], [17], [18] and [19]).When A(k) ...

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Power residues on Abelian varieties

Cited by 18 publications

References 8 publications

On the order of the reduction of a point on an abelian variety

On the order of the reduction of a point on an abelian variety

Local–global divisibility by 4 in elliptic curves defined over $${\mathbb{Q}}$$

Elliptic curves with {\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3) and counterexamples to local-global divisibility by 9

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