Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T04:47:24.749Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete addition laws on abelian varieties

Part of: Arithmetic algebraic geometry Finite fields and commutative rings (number-theoretic aspects) Abelian varieties and schemes

Published online by Cambridge University Press:  01 September 2012

Christophe Arene ,
David Kohel  and
Christophe Ritzenthaler
Christophe Arene
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: [email protected])
David Kohel
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: [email protected])
Christophe Ritzenthaler
Affiliation:
Institut de Mathématiques de Luminy, 163 Avenue de Luminy, Case 907, 13288 Marseille, France (email: [email protected])

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 prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

MSC classification

Secondary: 11G20: Curves over finite and local fields 14K15: Arithmetic ground fields 11T71: Algebraic coding theory; cryptography
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 308 - 316
Copyright
Copyright © London Mathematical Society 2012

References

[1]Arene, C. and Cosset, R., ‘Construction of a k-complete addition law on abelian surfaces’, Arithmetic, geometry, cryptography and coding theory 2011, Contemporary Mathematics 574 (American Mathematical Society, Providence, RI, 2012).Google Scholar
[2]Bernstein, D. J., Kohel, D. and Lange, T., ‘Twisted hessian curves’, Preprint, 2009.Google Scholar
[3]Bernstein, D. J. and Lange, T., ‘Faster addition and doubling on elliptic curves’, Advances in cryptology – ASIACRYPT 2007, Lecture Notes in Computer Science 4833 (Springer, Berlin, 2007) 29–50.CrossRefGoogle Scholar
[4]Bernstein, D. J. and Lange, T., ‘Complete addition laws for all elliptic curves over finite fields’, presentation, 2009, cr.yp.to/talks/2009.07.17/slides.pdf.Google Scholar
[5]Birkenhake, C. and Lange, H., Complex abelian varieties, 2nd edn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 302 (Springer, Berlin, 2004).Google Scholar
[6]Bosma, W. and Lenstra, H. W. Jr., ‘Complete systems of two addition laws for elliptic curves’, J. Number Theory 53 (1995) 229–240.CrossRefGoogle Scholar
[7]Farkas, H. M. and Kra, I., Riemann surfaces, 2nd edn, Graduate Texts in Mathematics 71 (Springer, New York, 1980).CrossRefGoogle Scholar
[8]Kohel, D., ‘Addition law structure of elliptic curves’, J. Number Theory 131 (2011) 894–919.CrossRefGoogle Scholar
[9]Lang, S., Abelian varieties (Springer, New York, 1983) reprint of the 1959 original.CrossRefGoogle Scholar
[10]Lang, S., Fundamentals of Diophantine geometry (Springer, New York, 1983).CrossRefGoogle Scholar
[11]Lange, H. and Ruppert, W., ‘Complete systems of addition laws on abelian varieties’, Invent. Math. 79 (1985) 603–610.CrossRefGoogle Scholar
[12]Lange, H. and Ruppert, W., ‘Addition laws on elliptic curves in arbitrary characteristics’, J. Algebra 107 (1987) 106–116.Google Scholar
[13]Mumford, D., ‘Varieties defined by quadratic equations’, Questions on algebraic varieties (C.I.M.E. summer school III, Ciclo, Varenna, 1969) (Edizioni Cremonese, Rome, 1970) 29–100.Google Scholar
[14]Mumford, D., Curves and their Jacobians (The University of Michigan Press, Ann Arbor, MI, 1975).Google Scholar
[15]Tsfasman, M. A. and Vlăduţ, S. G., Algebraic-geometric codes, Mathematics and its Applications (Soviet Series) 58 (Kluwer, Dordrecht, 1991) translated from the Russian by the authors.CrossRefGoogle Scholar