Papers › Explicit isogenies of prime degree over quadratic fields

Explicit isogenies of prime degree over quadratic fields

7 Jan 2021arXiv:2101.02673links table onlyarchive 2025-07-28

Barinder S. Banwait

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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 ℚ(√(-10)), ℚ(√(5)), and ℚ(√(7)), providing the first such examples after Mazur's 1978 determination for K = ℚ. The termination of the algorithm relies on the Generalised Riemann Hypothesis.

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