Doubly isogenous genus-2 curves with $D_4$-action
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- by Vishal Arul, Jeremy Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, Rachel Pries and Caleb Springer
- Math. Comp. 93 (2024), 347-381
- DOI: https://doi.org/10.1090/mcom/3891
- Published electronically: August 31, 2023
- HTML | PDF
Abstract:
We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose $C$ and $C’$ are curves over a finite field $K$, with $K$-rational base points $P$ and $P’$, and let $D$ and $D’$ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-$2$ maps on their Jacobians. We say that $(C,P)$ and $(C’,P’)$ are doubly isogenous if $Jac(C)$ and $Jac(C’)$ are isogenous over $K$ and $Jac(D)$ and $Jac(D’)$ are isogenous over $K$. For curves of genus $2$ whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.References
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Bibliographic Information
- Vishal Arul
- Affiliation: Unaffiliated mathematician, Sunnyvale, California 94086
- MR Author ID: 1322018
- ORCID: 0000-0002-4500-8968
- Email: varul.math@gmail.com
- Jeremy Booher
- Affiliation: School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
- MR Author ID: 912358
- ORCID: 0000-0003-1245-407X
- Email: jeremybooher@ufl.edu
- Steven R. Groen
- Affiliation: Department of Mathematics, Lehigh University, 17 Memorial Drive E, Bethlehem, Pennsylvania 18018
- MR Author ID: 1539878
- ORCID: 0000-0001-8595-4281
- Email: stg323@lehigh.edu
- Everett W. Howe
- Affiliation: Unaffiliated mathematician, San Diego, California 92104
- MR Author ID: 236352
- ORCID: 0000-0003-4850-8391
- Email: however@alumni.caltech.edu
- Wanlin Li
- Affiliation: Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130
- MR Author ID: 1278302
- Email: wanlin@wustl.edu
- Vlad Matei
- Affiliation: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest RO-70700, Romania
- MR Author ID: 979827
- ORCID: setImmediate$0.5682610860786037$2
- Email: vmatei@imar.ro
- Rachel Pries
- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 665775
- ORCID: 0000-0001-5987-0324
- Email: pries@colostate.edu
- Caleb Springer
- Affiliation: Department of Mathematics, University College London, Gower Street, London WC1H 0AY; and The Heilbronn Institute for Mathematical Research, Bristol, UK
- MR Author ID: 1326228
- ORCID: 0000-0003-1514-4755
- Email: c.springer@ucl.ac.uk
- Received by editor(s): February 25, 2021
- Received by editor(s) in revised form: January 27, 2023
- Published electronically: August 31, 2023
- Additional Notes: This work was supported by a grant from the Simons Foundation (546235) for the collaboration ‘Arithmetic Geometry, Number Theory, and Computation’, through a workshop held at ICERM. The second author was partially supported by the Marsden Fund Council administered by the Royal Society of New Zealand. The fifth author was partially funded by the Simons collaboration on ‘Arithmetic Geometry, Number Theory, and Computation’. The seventh author was partially supported by NSF grant DMS-19-01819. The eighth author was partially supported by National Science Foundation Awards CNS-2001470 and CNS-1617802.
- © Copyright 2023 by the authors
- Journal: Math. Comp. 93 (2024), 347-381
- MSC (2020): Primary 11G20, 11M38, 14H40, 14K02, 14Q05; Secondary 11G10, 11Y40, 14H25, 14H30, 14Q25
- DOI: https://doi.org/10.1090/mcom/3891
- MathSciNet review: 4654625