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Families of explicitly isogenous Jacobians of variable-separated curves

Part of: Computational number theory Arithmetic algebraic geometry Abelian varieties and schemes

Published online by Cambridge University Press:  01 August 2011

Benjamin Smith
Benjamin Smith*
Affiliation:
INRIA Saclay–Île-de-France, Laboratoire d’Informatique (LIX), École Polytechnique, 91128 Palaiseau Cedex, France (email: smith@lix.polytechnique.fr)

Abstract

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We construct six infinite series of families of pairs of curves (X,Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by 2, 3 or 4. For each family, we compute the isomorphism type of the isogeny kernel and the dimension of the image of the family in the appropriate moduli space. The families are derived from Cassou-Noguès and Couveignes’ explicit classification of pairs (f,g) of polynomials such that f(x1)−g(x2) is reducible.

Supplementary materials are available with this article.

MSC classification

Secondary: 14K02: Isogeny 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11Y99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 179 - 199
Copyright
Copyright © London Mathematical Society 2011

References

[1]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, vol. 1, Grundlehren der Mathematischen Wissenschaften 267 (Springer, Berlin, 1984).Google Scholar
[2]Avanzi, R. M., ‘A study on polynomials in separated variables with low genus factors’, PhD Thesis, Universität Essen, 2001.Google Scholar
[3]Birkenhake, Ch. and Lange, H., Complex abelian varieties, 2nd edn, Grundlehren der Mathematischen Wissenschaften 302 (Springer, Berlin, 2004).CrossRefGoogle Scholar
[4]Bosma, W. and Cannon, J. J., Handbook of magma functions (School of Mathematics and Statistics, University of Sydney, 1995).Google Scholar
[5]Bosma, W., Cannon, J. J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.CrossRefGoogle Scholar
[6]Bost, J.-B. and Mestre, J.-F., ‘Moyenne arithmético–géometrique et périodes des courbes de genre 1 et 2’, Gaz. Math. 38 (1988) 36–64.Google Scholar
[7]Brumer, A., ‘The rank of J 0(N)’, Astérisque 228 (1995) 41–68.Google Scholar
[8]Bruns, W. and Gubeladze, J., ‘Polytopal linear groups’, J. Algebra 218 (1999) 715–737.CrossRefGoogle Scholar
[9]Cassels, J. W. S., ‘Factorization of polynomials in several variables’, Proceedings of the 15th scandinavian congress, Oslo 1968, Lecture Notes in Mathematics 118 (Springer, New York, 1970) 1–17.Google Scholar
[10]Cassou-Noguès, P. and Couveignes, J.-M., ‘Factorisations explicities de g(y)−h(z)’, Acta Arith. 87 (1999) no. 4, 291–317.CrossRefGoogle Scholar
[11]Castryck, W. and Voight, J., ‘On nondegeneracy of curves’, Algebra Number Theory 3 (2009) no. 3, 255–281.CrossRefGoogle Scholar
[12]Chai, C.-L. and Oort, F., ‘A note on the existence of absolutely simple Jacobians’, J. Pure Appl. Algebra 155 (2001) no. 2–3, 115–120.CrossRefGoogle Scholar
[13]Davenport, H., Lewis, D. J. and Schinzel, A., ‘Equations of the form f(x)=g(y)’, Q. J. Math. Oxford 12 (1961) 304–312.CrossRefGoogle Scholar
[14]Davenport, H. and Schinzel, A., ‘Two problems concerning polynomials’, J. reine angew. Math. 214 (1964) 386–391.CrossRefGoogle Scholar
[15]Donagi, R. and Livné, R., ‘The arithmetic–geometric mean and isogenies for curves of higher genus’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999) no. 2, 323–339.Google Scholar
[16]Feit, W., ‘Automorphisms of symmetric balanced incomplete block designs’, Math. Z. 118 (1970) 40–49.CrossRefGoogle Scholar
[17]Feit, W., ‘On symmetric balanced incomplete block designs with doubly transitive automorphism groups’, J. Combin. Theory Ser. A 14 (1973) 221–247.CrossRefGoogle Scholar
[18]Feit, W., Some consequences of the classification of finite simple groups, Proceedings of Symposia in Pure Mathematics 37 (American Mathematical Society, Providence, RI, 1980) 175–181.Google Scholar
[19]Fried, M., Home page, http://www.math.uci.edu/∼mfried/.Google Scholar
[20]Fried, M., ‘On a conjecture of Schur’, Michigan Math. J. 17 (1970) 41–55.CrossRefGoogle Scholar
[21]Fried, M., ‘The field of definition of function fields and a problem in the reducibility of polynomials in two variables’, Illinois J. Math. 17 (1973) 128–146.CrossRefGoogle Scholar
[22]Fried, M., Exposition on an arithmetic–group theoretic connection via Riemann’s existence theorem, Proceedings of Symposia in Pure Mathematics 37 (American Mathematical Society, Providence, RI, 1980) 571–602.Google Scholar
[23]Fulton, W., Intersection theory, 2nd edn (Springer, Berlin, 1998).CrossRefGoogle Scholar
[24]Gaudry, P. and Gürel, N., ‘An extension of Kedlaya’s point-counting algorithm to superelliptic curves’, Advances in cryptology: ASIACRYPT 2001, Lecture Notes in Computer Science 2248 (ed. Boyd, C.; Springer, Berlin, 2001) 480–494.CrossRefGoogle Scholar
[25]Gorenstein, D., Lyons, R. and Solomon, R., The classification of the finite simple groups, Mathematical Surveys and Monographs 40.1 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
[26]Harrison, M. C., ‘Some notes on Kedlaya’s algorithm for hyperelliptic curves’, Preprint, 2010, arXiv math.NT/1006.4206 v1.Google Scholar
[27]Hashimoto, K.-I., ‘On Brumer’s family of RM-curves of genus two’, Tohoku Math. J. (2) 52 (2000) no. 4, 475–488.CrossRefGoogle Scholar
[28]Howe, E. W. and Zhu, H. J., ‘On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field’, J. Number Theory 92 (2002) 139–163.CrossRefGoogle Scholar
[29]Kedlaya, K. S., ‘Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology’, J. Ramanujan Math. Soc. 16 (2001) no. 4, 323–338.Google Scholar
[30]Koelman, R. J., ‘The number of moduli of families of curves on toric surfaces’, PhD Thesis, Catholic University, Nijmegen, 1991.Google Scholar
[31]Kux, G., ‘Construction of algebraic correspondences between hyperelliptic function fields using Deuring’s theory’, PhD Thesis, Universität Kaiserslautern, 2004.Google Scholar
[32]Lidl, R., Mullen, G. L. and Turnwald, G., Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics 65 (Longman Scientific and Technical, Harlow; copublished in the United States, Wiley, New York, 1993).Google Scholar
[33]Mestre, J.-F., ‘Couples de jacobiennes isogénes de courbes hyperelliptiques de genre arbitraire’, Preprint, 2009, arXiv:math.AG/0902.3470 v1.Google Scholar
[34]Mestre, J.-F., ‘Familles de courbes hyperelliptiques à multiplications réelles’, Arithmetic algebraic geometry (Texel, 1989), Progress in Mathematics 89 (Birkhäuser, Boston, MA, 1991).Google Scholar
[35]Oort, F. and Ueno, K., ‘Principally polarized abelian varieties of dimension two or three are Jacobian varieties’, J. Fac. Sci. Univ. Tokyo Sect. IA: Math. 20 (1973) 377–381.Google Scholar
[36]Poonen, B. and Schaefer, E. F., ‘Explicit descent for Jacobians of cyclic covers of the projective line’, J. reine angew. Math. 488 (1997) 141–188.Google Scholar
[37]Reid, M., ‘Graded rings and varieties in weighted projective space’, Manuscript,www.maths.warwick.ac.uk/∼miles/.Google Scholar
[38]Schaefer, E. F., ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998) 447–471.CrossRefGoogle Scholar
[39]Shimura, G., Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46 (Princeton University Press, Princeton, NJ, 1998).CrossRefGoogle Scholar
[40]Smith, B., ‘Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves’, EUROCRYPT 2008, Lecture Notes in Computer Science 4965 (ed. Smart, N.; Springer, Berlin, 2008) 163–180.CrossRefGoogle Scholar
[41]Smith, B., ‘Families of explicit isogenies of hyperelliptic Jacobians’, Arithmetic, geometry, cryptography and coding theory 2009, Contemporary Mathematics 521 (eds Kohel, D. and Rolland, R.; American Mathematical Society, Providence, RI, 2010) 121–144.CrossRefGoogle Scholar
[42]Tautz, W., Top, J. and Verberkmoes, A., ‘Explicit hyperelliptic curves with real multiplication and permutation polynomials’, Canad. J. Math. 43 (1991) no. 5, 1055–1064.CrossRefGoogle Scholar
[43]Vélu, J., ‘Isogénies entre courbes elliptiques’, C. R. Acad. Sci. Paris 273 (1971) 238–241.Google Scholar
[44]Zarhin, Yu. G., ‘Hyperelliptic Jacobians without complex multiplication, doubly transitive permutation groups and projective representations’, Algebraic number theory and algebraic geometry, Contemporary Mathematics 300 (eds Vostokov, S. and Zarhin, Y.; American Mathematical Society, Providence, RI, 2002) 195–210.CrossRefGoogle Scholar
[45]Zarhin, Yu. G., ‘The endomorphism rings of Jacobians of cyclic covers of the projective line’, Math. Proc. Cambridge Philos. Soc. 136 (2004) no. 2, 257–267.CrossRefGoogle Scholar
[46]Zarhin, Yu. G., ‘Superelliptic Jacobians’, Diophantine geometry, CRM Series 4 (Edizioni Della Normale, Pisa, 2007) 363–390.Google Scholar
[47]Zarhin, Yu. G., ‘Endomorphisms of superelliptic Jacobians’, Math. Z. 261 (2009) 691–707, 709.CrossRefGoogle Scholar
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