A local-global principle for isogenies of prime degree over number fields

Abstract

We give a description of the set of exceptional pairs for a number field $K$, that is the set of pairs $(\ell ,j(E))$, where $\ell$ is a prime and $j(E)$ is the $j$-invariant of an elliptic curve $E$ over $K$ which admits an $\ell$-isogeny locally almost everywhere but not globally. We obtain an upper bound for $\ell$ in such pairs in terms of the degree and the discriminant of $K$. Moreover, we prove finiteness results about the number of exceptional pairs.

22 pages, presentation improved as suggested by the referees. To appear in Journal of London Mathematical Society. arXiv admin note: text overlap with arXiv:1006.1782 by other authors

References in corpus (1)

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

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