Abstract
We give the explicit equations for a \({\textbf{P}}^3 \times {\textbf{P}}^3\) embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in \({\textbf{P}}^{15}\). We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law.
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The supplementary information file [10] includes details of the calculations performed using the computer algebra software Maple. The file is available from the website of the first-named author, as described in the references, and is also available as an ancillary file from arxiv:2211.01450. There is also a shortened version of this file [11] which gives only the assignments of the main objects (such as the diagonalising change in basis, the defining equations and the group law), which can be used in any algebra package.
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Both authors thank Michael Stoll for organizing the Rational Points 2019 workshop in Schney, where they began working together on this problem. The second named author (KKM) gratefully acknowledges generous funding and a supportive research environment during long scientific visits at both the Max Planck Institute for Mathematics in Bonn (2021) and the Institute for Advanced Study in Princeton (2022, with funding from the Charles Simonyi Endowment).
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Flynn, E.V., Khuri-Makdisi, K. An analog of the Edwards model for Jacobians of genus 2 curves. Res. number theory 10, 32 (2024). https://doi.org/10.1007/s40993-024-00518-5
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DOI: https://doi.org/10.1007/s40993-024-00518-5