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arXiv · 2101.02673

Explicit isogenies of prime degree over quadratic fields

Barinder S. Banwait
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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.

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Barinder S. Banwait. 2021-01-07. Explicit isogenies of prime degree over quadratic fields. https://doi.org/10.1093/imrn%2Frnac134

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