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An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three

In: Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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  • Michael Stoll

    (Universität Bayreuth, Mathematisches Institut)

Abstract

We develop an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field k of characteristic ≠ 2. In particular, we provide explicit equations defining the Kummer variety K $$\mathscr {K}$$ as a subvariety of , together with explicit polynomials giving the duplication map on . A careful study of the degenerations of this map then forms the basis for the development of an explicit theory of heights on such Jacobians when k is a number field. We use this input to obtain a good bound on the difference between naive and canonical height, which is a necessary ingredient for the explicit determination of the Mordell-Weil group. We illustrate our results with two examples.

Suggested Citation

  • Michael Stoll, 2017. "An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three," Springer Books, in: Gebhard Böckle & Wolfram Decker & Gunter Malle (ed.), Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, pages 665-715, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-70566-8_29
    DOI: 10.1007/978-3-319-70566-8_29
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