Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T05:04:09.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing functions on Jacobians and their quotients

Part of: Computational aspects in algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  01 August 2015

Jean-Marc Couveignes  and
Tony Ezome
Jean-Marc Couveignes
Affiliation:
Univ. Bordeaux, Bordeaux INP, CNRS, IMB, UMR 5251, F-33400 Talence, France INRIA, LFANT, LIRIMA, F-33400 Talence, France email [email protected]
Tony Ezome
Affiliation:
Université des Sciences et Techniques de Masuku, Faculté des Sciences, Département de mathématiques et informatique, BP 943 Franceville, Gabon INRIA, LIRIMA, F-33400 Talence, 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 show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute $(l,l)$ isogenies between Jacobians of genus two curves in quasi-linear time in the degree $l^{2}$.

MSC classification

Primary: 14K02: Isogeny 14Q05: Curves 14Q20: Effectivity, complexity
Secondary: 14K25: Theta functions
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 555 - 577
Copyright
© The Author(s) 2015 

References

Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, vol. I , Grundlehren der Mathematischen Wissenschaften 267 (Springer, New York, 1985).CrossRefGoogle Scholar
Baker, H. F., Abel’s theorem and the allied theory, including the theory of the theta functions (Cambridge University Press, Cambridge, 1897).Google Scholar
Birkenhake, C. and Lange, H., Complex abelian varieties , 2nd edn, Grundlehren der Mathematischen Wissenschaften 302 (Springer, Berlin, 2004).Google Scholar
Bostan, A., Morain, F., Salvy, B. and Schost, É., ‘Fast algorithms for computing isogenies between elliptic curves’, Math. Comp. 77 (2008) no. 263, 1755–1778.CrossRefGoogle Scholar
Brent, R. P. and Kung, H. T., ‘Fast algorithms for manipulating formal power series’, J. Assoc. Comput. Mach. 25 (1978) no. 4, 581–595.Google Scholar
Bürgisser, P., Clausen, M. and Amin Shokrollahi, M., Algebraic complexity theory , Grundlehren der Mathematischen Wissenschaften 315 (Springer, Berlin, 1997) ; with the collaboration of Thomas Lickteig.Google Scholar
Cosset, R. and Robert, D., ‘Computing (l, l)-isogenies in polynomial time on Jacobians of genus 2 curves’, Math. Comp. 84 (2015) 1953–1975.CrossRefGoogle Scholar
Couveignes, J.-M., ‘Linearizing torsion classes in the Picard group of algebraic curves over finite fields’, J. Algebra 321 (2009) no. 8, 2085–2118.Google Scholar
Couveignes, J.-M., ‘Computing V f modulo p ’, Computational aspects of modular forms and Galois representations , Annals of Mathematics Studies 176 (Princeton University Press, Princeton, NJ, 2011) 337–370.Google Scholar
Couveignes, J.-M. and Edixhoven, B., ‘First description of the algorithms’, Computational aspects of modular forms and Galois representations , Annals of Mathematics Studies 176 (Princeton University Press, Princeton, NJ, 2011) 69–78.CrossRefGoogle Scholar
Diem, C., ‘On arithmetic and the discrete logarithm problem in class groups of curves’, Habilitation Thesis, Leipzig, 2008.Google Scholar
Dolgachev, I. and Lehavi, D., ‘On isogenous principally polarized abelian surfaces’, Curves and abelian varieties , Contemporary Mathematics 465 (American Mathematical Society, Providence, RI, 2008) 51–69.Google Scholar
Elkies, N. D., ‘Elliptic and modular curves over finite fields and related computational issues’, Computational perspectives on number theory (Chicago, IL, 1995) , AMS/IP Studies in Advanced Mathematics 7 (American Mathematical Society, Providence, RI, 1998) 21–76.Google Scholar
Fay, J. D., Theta functions on Riemann surfaces , Lecture Notes in Mathematics 352 (Springer, Berlin, 1973).Google Scholar
Le Gall, F., ‘Powers of tensors and fast matrix multiplication’, ISSAC 2014 Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ACM, New York, 2014) 296–303.Google Scholar
Ghorpade, S. R. and Lachaud, G., ‘Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields’, Mosc. Math. J. 2 (2002) no. 3, 589–631; dedicated to Yuri I. Manin on the occasion of his 65th birthday.Google Scholar
Hess, F., ‘Computing Riemann–Roch spaces in algebraic function fields and related topics’, J. Symbolic Comput. 33 (2002) no. 4, 425–445.CrossRefGoogle Scholar
Joux, A. and Lercier, R., ‘Counting points on elliptic curves in medium characteristic’. Cryptology ePrint Archive, Report 2006/176, 2006.Google Scholar
Khuri-Makdisi, K., ‘Asymptotically fast group operations on Jacobians of general curves’, Math. Comp. 76 (2007) no. 260, 2213–2239.Google Scholar
Lachaud, G. and Martin-Deschamps, M., ‘Nombre de points des jacobiennes sur un corps fini’, Acta Arith. 56 (1990) no. 4, 329–340.Google Scholar
Lang, S., Abelian varieties (Springer, New York, Berlin, 1983) ; reprint of the 1959 original.Google Scholar
Lercier, R. and Morain, F., ‘Algorithms for computing isogenies between elliptic curves’, Computational perspectives on number theory: proceedings of a conference in honor of A. O. L. Atkin , AMS/IP Studies in Advanced Mathematics 7 (eds Buell, D. A. and Teitelbaum, J. T.; American Mathematical Society & International Press, Providence, RI, 1998) 77–96. Held in 1995 at the University of Illinois at Chicago.Google Scholar
Lubicz, D. and Robert, D., ‘Computing isogenies between abelian varieties’, Compos. Math. 148 (2012) no. 5, 1483–1515.CrossRefGoogle Scholar
Lubicz, D. and Robert, D., ‘Computing separable isogenies in quasi-optimal time’, LMS J. Comput. Math. 18 (2015) no. 1, 198–216.Google Scholar
Mascot, N., ‘Computing modular Galois representations’, Rend. Circ. Mat. Palermo (2) 62 (2013) no. 3, 451–476.Google Scholar
Miller, V. S., ‘The Weil pairing, and its efficient calculation’, J. Cryptology 17 (2004) no. 4, 235–261.CrossRefGoogle Scholar
Milne, J. S., ‘Abelian varieties’, Arithmetic geometry (Springer, New York, 1986) 103–150.Google Scholar
Mumford, D., Tata lectures on theta. III, Modern Birkhäuser classics (Birkhäuser Boston, Inc., Boston, MA, 2007) ; with collaboration of Madhav Nori and Peter Norman. Reprint of the 1991 original.Google Scholar
D. Mumford  Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5 (Published for the Tata Institute of Fundamental Research, Bombay, 2008); with appendices by C. P. Ramanujam and Y. Manin. Corrected reprint of the second 1974 edition.Google Scholar
Schoof, R., ‘Counting points on elliptic curves over finite fields’, J. Théor. Nombres Bordeaux 7 (1995) no. 1, 219–254; Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993).CrossRefGoogle Scholar
Shepherd-Barron, N., ‘Thomae’s formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves’, Preprint, 2008, arXiv:0802.3014.Google Scholar
Smith, B., ‘Computing low-degree isogenies in genus 2 with the Dolgachev–Lehavi method’, Arithmetic, geometry, cryptography and coding theory , Contemporary Mathematics 574 (American Mathematical Society, Providence, RI, 2012) 159–170.Google Scholar
The PARI Group, Bordeaux. PARI/GP version 2.7.2, 2014, available fromhttp://pari.math.u-bordeaux.fr/.Google Scholar
Vélu, J., ‘Isogénies entre courbes elliptiques’, C. R. Acad. Sci. Paris 273 (1971) A238–A241.Google Scholar
Vélu, J., ‘Courbes elliptiques munies d’un sous-groupe Z∕n Z ×𝜇 n ’, Bull. Soc. Math. France Mém. 57 (1978) 5–152.Google Scholar
Weil, A., ‘Sur les courbes algébriques et les variétés qui s’en déduisent’, Actualités Sci. Ind., no. 1041 , Publ. Inst. Math. Univ. Strasbourg 7, 1945 (Hermann et Cie, Paris, 1948).Google Scholar
Weil, A., ‘Variétés abéliennes et courbes algébriques’, Actualités Sci. Ind., no. 1064 , Publ. Inst. Math. Univ. Strasbourg 8, 1946 (Hermann & Cie, Paris, 1948).Google Scholar