An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three

Michael Stoll ()
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Michael Stoll: Universität Bayreuth, Mathematisches Institut

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 665-715 from Springer

Abstract: 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.

Keywords: Kummer variety; Hyperelliptic curve; Genus 3; Canonical height; 14H40; 14H45; 11G10; 11G50; 14Q05; 14Q15 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-70566-8_29

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DOI: 10.1007/978-3-319-70566-8_29

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