Explicit isogenies of prime degree over quadratic fields

TL;DR

This paper develops an algorithm to identify primes for which elliptic curves over certain quadratic fields have rational p-isogenies, extending previous work and providing explicit examples beyond the rational case.

Contribution

It introduces a new algorithm for computing primes with rational p-isogenies over quadratic fields, and applies it to three specific fields, expanding known cases beyond Mazur's classical results.

Findings

Determined the set of primes with rational p-isogenies for three quadratic fields.

Provided explicit examples of such primes beyond the rational case.

The algorithm's termination depends on the Generalised Riemann Hypothesis.

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, \"{O}zman-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.