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Explicit Kummer surface formulas for arbitrary characteristic

Part of: Computational aspects in algebraic geometry Arithmetic algebraic geometry Finite fields and commutative rings (number-theoretic aspects) Curves Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 January 2010

Jan Steffen Müller
Jan Steffen Müller*
Affiliation:
Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany (email: [email protected])

Abstract

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If C is a curve of genus 2 defined over a field k and J is its Jacobian, then we can associate a hypersurface K in ℙ3 to J, called the Kummer surface of J. Flynn has made this construction explicit in the case when the characteristic of k is not 2 and C is given by a simplified equation. He has also given explicit versions of several maps defined on the Kummer surface and shown how to perform arithmetic on J using these maps. In this paper we generalize these results to the case of arbitrary characteristic.

MSC classification

Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14H40: Jacobians, Prym varieties 14Q10: Surfaces, hypersurfaces 11T71: Algebraic coding theory; cryptography 11G50: Heights
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 47 - 64
Copyright
Copyright © London Mathematical Society 2010

References

[1] 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
[2] Duquesne, S., ‘Traces of the group law on the Kummer surface of a curve of genus 2 in characteristic 2’, Preprint, 2007, http://perso.univ-rennes1.fr/sylvain.duquesne/Google Scholar
[3] Duquesne, S., ‘Montgomery scalar multiplication for genus 2 curves’, ANTS VI, Burlington, VT, 2004, Lecture Notes in Computer Science, 3076 (ed. Buell, D. A.; Springer, Berlin, 2004) 153–168.Google Scholar
[4] Duquesne, S., ‘Montgomery ladder for all genus 2 curves in characteristic 2’, WAIFI 2008, Lecture Notes in Computer Science, 5130 (Springer, Berlin, 2008) 174–188.Google Scholar
[5] 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) 425–441.CrossRefGoogle Scholar
[6] Flynn, E. V., ‘The group law on the jacobian of a curve of genus 2’, J. reine angew. Math. 439 (1995) 45–69.Google Scholar
[7] 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) 333–352.CrossRefGoogle Scholar
[8] Gaudry, P., ‘Fast genus 2 arithmetic based on Theta functions’, J. Math. Crypt. 1 (2007) 243–265.Google Scholar
[9] Gaudry, P. and Lubicz, D., ‘The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines’, Finite Fields Theory Appl. 15 (2009) 246–260.CrossRefGoogle Scholar
[10] Hudson, R. W. H. T., Kummer’s Quartic Surface (University Press, Cambridge, 1905).Google Scholar
[11] Lang, S., Introduction to Algebraic and Abelian Functions, 2nd edn (Springer, New York, 1982).CrossRefGoogle Scholar
[12] Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic Comput. 24 (1997) 235–265. (See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.)CrossRefGoogle Scholar
[13] Monagan, M. B., Geddes, K. O., Heal, K. M., Labahn, G., Vorkoetter, S. M., McCarron, J. and DeMarco, P., Maple 10 Programming Guide (Maplesoft, Waterloo ON, Canada, 2005). (See also the Maple homepage at http://www.maplesoft.com/.)Google Scholar
[14] Stoll, M., ‘On the height constant for curves of genus two, II’, Acta Arith. 104 (2002) 165–182.CrossRefGoogle Scholar
Supplementary material: File

Müller supplementary material

Formulae appendix

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