Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over Q \mathbb {Q} . Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) with | d | > 10 4 |d| > 10^4 we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over 19 19 quadratic fields, including Q ( 213 ) \mathbb {Q}(\sqrt {213}) and Q ( − 2289 ) \mathbb {Q}(\sqrt {-2289}) . To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X 0 ( 125 ) X_0(125) and X 0 ( 169 ) X_0(169) , which may be of independent interest.