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Explicit rational group law on hyperelliptic Jacobians of any genus

Volume 71 / 2023

David Urbanik Bulletin Polish Acad. Sci. Math. 71 (2023), 35-52 MSC: Primary 14H40; Secondary 14K05. DOI: 10.4064/ba230330-7-7 Published online: 19 July 2023


It is well-known that abelian varieties are projective, and so there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford’s classic “Lectures on Theta”, where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.


  • David UrbanikInstitut des Hautes Études Scientifiques
    91440 Paris, France

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