Cyclic isogenies of elliptic curves over fixed quadratic fields
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- by Barinder S. Banwait, Filip Najman and Oana Padurariu;
- Math. Comp. 93 (2024), 841-862
- DOI: https://doi.org/10.1090/mcom/3894
- Published electronically: August 30, 2023
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Abstract:
Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb {Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised.
In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb {Q}(\sqrt {d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb {Q}(\sqrt {213})$ and $\mathbb {Q}(\sqrt {-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.
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Bibliographic Information
- Barinder S. Banwait
- Affiliation: Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, Massachusetts 02215
- MR Author ID: 1079808
- Email: barinder.s.banwait@gmail.com
- Filip Najman
- Affiliation: University of Zagreb, Bijenička Cesta 30, 10000 Zagreb, Croatia
- MR Author ID: 886852
- ORCID: 0000-0002-0994-0846
- Email: fnajman@math.hr
- Oana Padurariu
- Affiliation: Department of Mathematics & Statistics, Boston University, 665 Commonwealth Avenue, Boston, Massachusetts 02215
- MR Author ID: 1529936
- Email: oana@bu.edu
- Received by editor(s): November 3, 2022
- Received by editor(s) in revised form: June 7, 2023
- Published electronically: August 30, 2023
- Additional Notes: The second author was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004) and by the Croatian Science Foundation under the project no. IP-2018-01-1313. The third author was supported by NSF grant DMS-1945452 and Simons Foundation grant #550023.
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 841-862
- MSC (2020): Primary 11G05; Secondary 11Y60, 11G15
- DOI: https://doi.org/10.1090/mcom/3894
- MathSciNet review: 4678586