Papers › Cyclic isogenies of elliptic curves over fixed quadratic fields
Cyclic isogenies of elliptic curves over fixed quadratic fields
Barinder S. Banwait, Filip Najman, Oana Padurariu
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Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over 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)) with |d| < 10⁴ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including ℚ(√(213)) and ℚ(√(-2289)). To make this procedure work, we determine all of the finitely many quadratic points on the modular curves X₀(125) and X₀(169), which may be of independent interest.
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