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An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three

  • Michael Stoll
Chapter

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 \(\mathscr {K}\) as a subvariety of Open image in new window , together with explicit polynomials giving the duplication map on Open image in new window . 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.

Keywords

Kummer variety Hyperelliptic curve Genus 3 Canonical height 

Subject Classifications:

14H40 14H45 11G10 11G50 14Q05 14Q15 

Notes

Acknowledgements

I would like to thank Steffen Müller for helpful comments on a draft version of this paper and for pointers to the literature. The necessary computations were performed using the Magma computer algebra system [1].

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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