A local-global principle for rational isogenies of prime degree

Details A local-global principle for rational isogenies of prime degree A local-global principle for rational isogenies of prime degree

2010-06-09

arxiv.org/abs/1006.1782v4

Mathematics

Number Theory

11 pages, minor edits, to appear in Journal de Th\'eorie des Nombres de Bordeaux

Scientific paper

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.

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