Rational points on X_0^+ (p^r)

Bilu, Yu.; Parent, P.; Rebolledo, M.

doi:10.48550/arxiv.1104.4641

Cited by 2 publications

(3 citation statements)

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“…For an element h ∈ H of order n, Proposition 2 implies that s = 2 + (ℓ − 1)/n is an integer, thus n divides ℓ − 1 and in fact divides (ℓ − 1)/2. This completes the proof of (1).…”

Section: Proof Of Theoremsupporting

confidence: 62%

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A local-global principle for rational isogenies of prime degree

Sutherland¹

2012

J. Théor. Nombres Bordeaux

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Let K be a number field. We consider a local-global principle for elliptic curves E/K that admit (or do not admit) a rational isogeny of prime degree n. For suitable K (including K=Q), we prove that this principle holds when n = 1 mod 4, and for n < 7, but find a counterexample when n = 7 for an elliptic curve with j-invariant 2268945/128. For K = Q we show that, up to isomorphism, this is the only counterexample.Comment: 11 pages, minor edits, to appear in Journal de Th\'eorie des Nombres de Bordeau

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Section: Proof Of Theoremsupporting

confidence: 62%

“…By Theorem 1.1 of [12], for all ℓ > 7 congruent to 3 mod 4, the only rational non-cuspidal points on X + 0 (ℓ 2 )(Q) correspond to elliptic curves with CM. 1 But we have shown that no curve with CM can arise in an exceptional pair, and the theorem follows.…”

Section: Proof Of Theoremmentioning

confidence: 70%

A local-global principle for rational isogenies of prime degree

Sutherland¹

2012

J. Théor. Nombres Bordeaux

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“…Le théorème de Mazur constitue, pour le corps Q, une partie de la réponse au ≪ problème de Serre uniforme ≫ (voir la partie 4.3 de [Ser72]) qui consiste à déterminer s'il existe une borne B K vérifiant : pour toute courbe elliptique E définie sur K et sans multiplication complexe sur Q et tout nombre premier p strictement supérieur à B K , la représentation ϕ E,p est surjective. On pourra à ce sujet consulter les travaux récents de Bilu, Parent et Rebolledo (voir [BPR11]).…”

Section: Introductionunclassified

Critère d'irréductibilité pour les courbes elliptiques semi-stables sur un corps de nombres

David

2012

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For a fixed number field and an elliptic curve defined and semi-stable over this number field, we consider the set of prime numbers p such that the Galois representation attached to the p-torsion points of the elliptic curve is reducible. When the number field satisfies a certain necessary condition, we give an explicit bound, depending only on the number field and not on the semi-stable elliptic curve, for these primes. This generalizes previous results of Kraus.

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Rational points on X_0^+ (p^r)

Cited by 2 publications

References 0 publications

A local-global principle for rational isogenies of prime degree

A local-global principle for rational isogenies of prime degree

Critère d'irréductibilité pour les courbes elliptiques semi-stables sur un corps de nombres

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