A Local-global principle for isogenies of composite degree

Abstract

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.

arXiv comment30 pages

References in corpus (5)

• Modular curves of prime-power level with infinitely many rational points
• A local-global principle for rational isogenies of prime degree
• Torsion points and Galois representations on CM elliptic curves
• Tetrahedral Elliptic Curves and the local-global principle for Isogenies
• A local-global principle for isogenies of prime degree over number fields

Cited by in corpus (3)

• On the local-global principle for isogenies of abelian surfaces
• Rational points on conic bundles over elliptic curves
• On Ribet's Lemma for ${\mathrm{GL}}_2$ modulo prime powers