Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication
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- by Bogdan Dina, Sorina Ionica and Jeroen Sijsling;
- Math. Comp. 92 (2023), 349-383
- DOI: https://doi.org/10.1090/mcom/3776
- Published electronically: August 22, 2022
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Abstract:
We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields $K$ for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian has complex multiplication by $\mathbb {Z}_K$. More precisely, we find all sextic CM fields $K$ in the LMFDB for which (heuristically) Jacobians of both types with CM by $\mathbb {Z}_K$ exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. We determine invariants of the corresponding curves, and in the simplest case we also give an explicit defining equation.References
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Bibliographic Information
- Bogdan Dina
- Affiliation: Eistein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel; Institut für Algebra und Zahlentheorie, Universität Ulm, Helmholtzstrasse 18, D-89081 Ulm, Germany
- MR Author ID: 1476109
- Email: bogdan.dina@mail.huji.ac.il
- Sorina Ionica
- Affiliation: Laboratoire MIS, Universite de Picardie, 33 Rue St-Leu, 80039 Amiens Cedex 1, France
- MR Author ID: 891818
- Email: sorina.ionica@u-picardie.fr
- Jeroen Sijsling
- Affiliation: Institut für Algebra und Zahlentheorie, Universität Ulm, Helmholtzstrasse 18, D-89081 Ulm, Germany
- MR Author ID: 974789
- ORCID: 0000-0002-0632-9910
- Email: jeroen.sijsling@uni-ulm.de
- Received by editor(s): May 18, 2021
- Received by editor(s) in revised form: February 27, 2022, and July 1, 2022
- Published electronically: August 22, 2022
- Additional Notes: The first and second authors were financially supported by the Thomas Jefferson Fund program, which was supported by the Embassy of France in the United States and the FACE Foundation. The third author was supported by the Juniorprofessuren-Programm “Endomorphismen algebraischer Kurven” of the Science Ministry of Baden-Württemberg as well as by a Sachbeihilfe “Abstieg algebraischer Kurven” of the Deutsche Forschungsgemeinschaft.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 349-383
- MSC (2020): Primary 14H40, 14H25; Secondary 14H45, 14H50, 14K20, 14K22
- DOI: https://doi.org/10.1090/mcom/3776
- MathSciNet review: 4496968