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Constructing genus-3 hyperelliptic Jacobians with CM

  • Jennifer S. Balakrishnan (a1), Sorina Ionica (a2), Kristin Lauter (a3) and Christelle Vincent (a4)

Abstract

Given a sextic CM field $K$ , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$ , and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field  $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo  $p$ .

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References

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1. Balakrishnan, J. S., Ionica, S., Lauter, K. and Vincent, C., ‘Genus 3’, 2016, https://github.com/christellevincent/genus3.
2. Birkenhake, C. and Lange, H., Complex abelian varieties , 2nd edn, Grundlehren der Mathematischen Wissenschaften 302 (Springer, Berlin, 2004).
3. Cohen, H., A course in computational algebraic number theory , Graduate Texts in Mathematics 138 (Springer, Berlin, 1993).
4. Cosset, R., ‘Applications des fonctions thêta à la cryptographie sur les courbes hyperelliptiques’, PhD Thesis, Université Henri Poincaré – Nancy I, 2011.
5. Costello, C., Deines-Schartz, A., Lauter, K. and Yang, T., ‘Constructing abelian surfaces for cryptography via Rosenhain invariants’, LMS J. Comput. Math. 17 (2014) no. A, 157180.
6. Diem, C., ‘An index calculus algorithm for plane curves of small degree’, Algorithmic number theory: 7th international symposium, ANTS VII , Lecture Notes in Computational Science 4076 (eds Hess, F., Pauli, S. and Pohst, M. E.; Springer, Berlin, 2006).
7. Dupont, R., ‘Moyenne arithmético-géométrique, suites de Borchardt et applications’, PhD Thesis, École Polytechnique, 2006.
8. Gaudry, P., Thomé, E., Thériault, N. and Diem, C., ‘A double large prime variation for small genus hyperelliptic index calculus’, Math. Comp. 76 (2007) 475492.
9. Gottschling, E., ‘Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades’, Math. Ann. 138 (1959) 103124.
10. Gross, B. and Harris, J., On some geometric constructions related to theta characteristics (Johns Hopkins University Press, Baltimore, MD, 2004) 279311.
11. Igusa, J., ‘Modular forms and projective invariants’, Amer. J. Math. 89 (1967) 817855.
12. Igusa, J., Theta functions , Grundlehren der mathematischen Wissenschaften 194 (Springer, New York, 1972).
13. Koike, K. and Weng, A., ‘Construction of CM Picard curves’, Math. Comp. 74 (2005) no. 249, 499518.
14. Laine, K. and Lauter, K., ‘Time-memory trade-offs for index calculus in genus 3’, J. Math. Cryptol. 9 (2015) no. 2, 95114.
15. Lang, S., Complex multiplication , Grundlehren der Mathematischen Wissenschaften 255 (Springer, New York, 1983).
16. Mumford, D., Tata lectures on theta. I , Modern Birkhäuser classics (Birkhäuser, Boston, 2007), with the collaboration of C. Musili, M. Nori, E. Previato and M. Stillman, Reprint of the 1983 edition.
17. Mumford, D., Tata lectures on theta. II , Modern Birkhäuser classics (Birkhäuser, Boston, 2007) with the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Reprint of the 1984 original.
18. Poor, C., ‘The hyperelliptic locus’, Duke Math. J. 76 (1994) no. 3, 809884.
19. Shimura, G. and Taniyama, Y., Complex multiplication of abelian varieties and its applications to number theory , Publications of the Mathematical Society of Japan 6 (Mathematical Society of Japan, Tokyo, 1961).
20. Smith, B., ‘Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves’, J. Cryptology 22 (2009) no. 4, 505529.
21. Spallek, A.-M., ‘Kurven von Geschlecht 2 und ihre Anwendung in Public Key Kryptosystemen’, PhD Thesis, Institut für Experimentelle Mathematik, Universität GH Essen, 1994.
22. Stein, W. A. et al. , ‘Sage Mathematics Software (Version 6.10)’, The Sage Development Team, 2015, http://www.sagemath.org.
23. Streng, M., ‘Complex multiplication of abelian surfaces’, PhD Thesis, Universiteit Leiden, 2010.
24. Takase, K., ‘A generalization of Rosenhain’s normal form for hyperelliptic curves with an application’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) no. 7, 162165.
25. van Wamelen, P., ‘Examples of genus two CM curves defined over the rationals’, Math. Comp. 68 (1999) no. 225, 307320.
26. Weber, H.-J., ‘Hyperelliptic simple factors of J 0(N) with dimension at least 3’, Experiment. Math. 6 (1997) no. 4, 273287.
27. Weng, A., ‘A class of hyperelliptic CM-curves of genus three’, J. Ramanujan Math. Soc. 16 (2001) no. 4, 339372.
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