Explicit isogenies of prime degree over quadratic fields

Abstract

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin-Najman, Özman-Siksek, and most recently Box, we determine the above set of primes for the three quadratic fields $\mathbb{Q}(\sqrt{-10})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{7})$, providing the first such examples after Mazur's 1978 determination for $K=\mathbb{Q}$. The termination of the algorithm relies on the Generalised Riemann Hypothesis.

36 pages, with an Appendix written jointly with Maarten Derickx. Manuscript changed after several rounds of Referee reports

Cited by in corpus (3)

• On the effective version of Serre's open image theorem
• On elliptic curves with $p$-isogenies over quadratic fields
• Explicit isogenies of prime degree over number fields