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Isogenies of non-CM elliptic curves with rational j-invariants over number fields

Published online by Cambridge University Press:  01 March 2017

FILIP NAJMAN
FILIP NAJMAN*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia. e-mail: fnajman@math.hr
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Abstract

We unconditionally determine $I_{\mathbb Q}(d)$, the set of possible prime degrees of cyclic K-isogenies of elliptic curves with ${\mathbb Q}$-rational j-invariants and without complex multiplication over number fields K of degree ≤ d, for d ≤ 7, and give an upper bound for $I_{\mathbb Q}(d)$ for d > 7. Assuming Serre's uniformity conjecture, we determine $I_{\mathbb Q}(d)$ exactly for all positive integers d.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 179 - 184
Copyright
Copyright © Cambridge Philosophical Society 2017 

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