A local-global principle for rational isogenies of prime degree
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 475-485.

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 . For suitable K (including K=), we prove that this principle holds for all 1mod4, and for <7, but find a counterexample when =7 for an elliptic curve with j-invariant 2268945/128. For K= we show that, up to isomorphism, this is the only counterexample.

Soit K un corps de nombres. Nous étudions un principe local-global pour les courbes elliptiques E/K admettant ou non une isogénie rationnelle de degré premier . Pour des corps K convenables (dont K=), nous démontrons ce principe pour tout 1mod4 et tout <7 mais exhibons une courbe elliptique d’invariant modulaire 2268945/128 comme contre-exemple pour =7. Nous montrons alors qu’il s’agit du seul contre-exemple à isomorphisme près lorsque K=.

DOI: 10.5802/jtnb.807
Classification: 11G05
Keywords: elliptic curve, isogeny, local-global principle
Sutherland, Andrew V. 1

1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139
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Sutherland, Andrew V. A local-global principle for rational isogenies of prime degree. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 475-485. doi : 10.5802/jtnb.807. http://www.numdam.org/articles/10.5802/jtnb.807/

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