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- Title
Explicit Isogenies of Prime Degree Over Quadratic Fields.
- Authors
Banwait, Barinder S
- 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.
- Subjects
RIEMANN hypothesis; QUADRATIC fields; ELLIPTIC curves
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 14, p11829
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac134