Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T10:39:45.739Z Has data issue: false hasContentIssue false

Explicit Kummer varieties of hyperelliptic Jacobian threefolds

Published online by Cambridge University Press:  01 September 2014

J. Steffen Müller*
Affiliation:
Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany email jan.steffen.mueller@uni-oldenburg.de

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a rational Weierstrass point defined over the same field. We also construct homogeneous quartic polynomials on the Kummer variety and show that they represent the duplication map using results of Stoll.

Supplementary materials are available with this article.

Type
Research Article
Copyright
© The Author 2014 

References

Birkenhake, C. and Lange, H., Complex Abelian varieties , 2nd edn (Springer, Berlin, 2004).Google Scholar
Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2 (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
Duquesne, S., ‘Calculs effectifs des points entier et rationnels sur les courbes’, Thèse de doctorat, Université Bordeaux I, 2001.Google Scholar
Duquesne, S., ‘Traces of the group law on the Kummer surface of a curve of genus 2 in characteristic 2’, Math. Comput. Sci. 3 (2010) 173183.Google Scholar
Flynn, E. V., ‘The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field’, Math. Proc. Camb. Phil. Soc. 107 (1990) 425441.Google Scholar
Flynn, E. V., ‘The group law on the Jacobian of a curve of genus 2’, J. reine angew. Math. 439 (1993) 4569.Google Scholar
Flynn, E. V. and Smart, N. P., ‘Canonical heights on the Jacobians of curves of genus 2 and the infinite descent’, Acta Arith. 79 (1997) 333352.Google Scholar
Holmes, D., ‘Computing Néron–Tate heights of points on hyperelliptic Jacobians’, J. Number Theory 132 (2012) no. 2, 12951305.Google Scholar
Holmes, D., ‘An Arakelov-Theoretic Approach to Naive Heights on Hyperelliptic Jacobians’, Preprint, 2012, arXiv:math/1207.5948v2 [math.NT].Google Scholar
Hudson, R. W. H. T., Kummer’s quartic surface (University Press, Cambridge, 1905).Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic. Comput. 24 (1997) 235265.Google Scholar
Müller, J. S., ‘Explicit Kummer surface formulas for arbitrary characteristic’, LMS J. Comput. Math. 13 (2010) 4764.Google Scholar
Müller, J. S., ‘Computing canonical heights on Jacobians’, PhD Thesis, Universität Bayreuth, 2010.Google Scholar
Müller, J. S., ‘Computing canonical heights using arithmetic intersection theory’, Math. Comput. 83 (2014) 311336.Google Scholar
Müller, J. S. and Stoll, M., ‘Canonical heights on Jacobians of genus two curves’, 2014 (in preparation).Google Scholar
Mumford, D., ‘On the equations defining abelian varieties. I’, Invent. Math. 1 (1966) 287354.Google Scholar
Mumford, D., Abelian varieties , Tata Institute of Fundamental Research Studies in Mathematics 5 (Tata Institute of Fundamental Research, Bombay, 1974).Google Scholar
Stoll, M., ‘On the height constant for curves of genus two’, Acta Arith. 90 (1999) 183201.Google Scholar
Stoll, M., ‘On the height constant for curves of genus two, II’, Acta Arith. 104 (2002) 165182.Google Scholar
Stoll, M., An explicit theory of heights for hyperelliptic Jacobians of genus three, 2014 (in preparation) (See also http://www.mathe2.uni-bayreuth.de/stoll/talks/Luminy2012.pdf.).Google Scholar
Stubbs, A. G. J., ‘Hyperelliptic curves’, PhD Thesis, University of Liverpool, 2000.Google Scholar
Supplementary material: File

Müller Supplementary Material

Supplementary Material

Download Müller Supplementary Material(File)
File 487.5 KB