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Ranks of Elliptic Curves Over Function Fields

Published online by Cambridge University Press:  01 February 2010

Alan G. B. Lauder
Alan G. B. Lauder
Affiliation:
Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, United Kingdom, lauder@maths.ox.ac.uk

Abstract

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We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology.

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008, pp. 172 - 212
Copyright
Copyright © London Mathematical Society 2008

References

1.Abbot, T.G., Kedlaya, K.S. and Roe, D., ‘Bounding Picard numbers of surfaces using p-adic cohomology’, Arithmetic, Geometry and Coding Theory (AGCT 2005), Société Mathématique de France, to appear. Available at arxiv:math/0601508v2.Google Scholar
2.Bektemirov, B., Mazur, B., Stein, W. and Watkins, M., ‘Average ranks of elliptic curves: tension between data and conjecture’, Bull. Amer. Math. Soc. 44 (2007) 233–254.CrossRefGoogle Scholar
3.Bolibrukh, A. A., ‘The Riemann-Hilbert problem’, Russian Math. Surveys 45(2) (1990) 1–47.CrossRefGoogle Scholar
4.Cox, D. A. and Katz, S., Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
5.Cox, D., Little, J. and O'Shea, D., Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, 2nd edn (Springer, New York, 1997).Google Scholar
6.Dekkers, W., ‘The matrix of a connection having regular singularities on a vector bundle of rank 2 on P 1(C)’, Lecture Notes in Mathematics 712 (Springer, Berlin, 1979) 33–43.Google Scholar
7.Deligne, P., ‘La conjecture de Weil: II’, Publ. Inst. Hautes Études Sci. 52 (1981) 313–428.Google Scholar
8.Dimca, A., ‘Singularities and coverings of weighted complete intersections’, J. reine angew. Math. 366 (1986) 184–193.Google Scholar
9.Dimca, A., ‘Singularities and topology of hypersurfaces’ (Springer, Berlin, 1992).CrossRefGoogle Scholar
10.Dimca, A., Sheaves in topology (Springer, Berlin, 2004).CrossRefGoogle Scholar
11.Gerkmann, R., ‘Relative rigid cohomology and deformation of hypersur faces’, Int. Math. Res. Pap. 2007 (2007), article ID rpm003, 67 pages.Google Scholar
12.Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics 52 (Springer, New York, 1977).CrossRefGoogle Scholar
13.Katz, N. M., ‘Nilpotent connections and the monodromy theorem: applications of a result of Turrittin’, Publ. Inst. Hautes Études Sci 39 (1970) 175–232.CrossRefGoogle Scholar
14.Katz, N. M., Twisted L-functions and monodromy, Annals of Mathematics Studies 150, (Princeton University Press, Princeton, NJ, 2002).Google Scholar
15.Katz, N. M., Moments, monodromy, and perversity: a diophantine perspective, Annals of Mathematics Studies 159 (Princeton University Press, Princeton, NJ, 2005).Google Scholar
16.Kedlaya, K. S., ‘Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology’, J. Ramanujan Math. Soc. 16 (2001) 323–338.Google Scholar
17.Kedlaya, K. S., ‘Finiteness of rigid cohomology with coefficients’, Duke Math. J. 134 (2006) 15–97.CrossRefGoogle Scholar
18.Kedlaya, K. S., ‘Search techniques for root-unitary polynomials’, Proceedings of a Special Session on Computational Arithmetic Geometry (Lauter, K. and Ribet, K., eds), American Mathematican Society, to appear. Available at arxiv:math/0611835v3.Google Scholar
19.Lauder, A. G. B., ‘A recursive method for computing zeta functions of varieties’, LMS J. Comput. Math. 9 (2006) 222–267.CrossRefGoogle Scholar
20.Lauder, A. G. B. and Wan, D., ‘Counting points on varieties over finite fields of small characteristic’, Algorithmic number theory: lattices, number fields, curves and cryptography(ed. Buhler, J. P. and Stevenhagen, P.), Math. Sci. Res. Inst. Publ. 44, to appear. Available at arxiv:math/0612147.Google Scholar
21.Le Stum, B., Rigid cohomology, Cambridge Tracts in Mathematics 172 (Cambridge University Press, Cambridge, 2007).CrossRefGoogle Scholar
22.Mazur, B., ‘Frobenius and the Hodge filtration’, Bull. Amer. Math. Soc. 78 (1972) 653–667.CrossRefGoogle Scholar
23.Milne, J. S., Etale cohomology, Princeton Mathematical Series (Princeton University Press, Princeton, 1980).CrossRefGoogle Scholar
24.Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer, New York, 1994).CrossRefGoogle Scholar
25.Steenbrink, J., ‘Intersection form for quasi-homogeneous singularities’, Compositio Math. 34 (1977) 211–223.Google Scholar
26.Tate, J., ‘Algebraic cycles and poles of zeta functions’, Arithmetic algebraic geometry, Proc. Conf. Purdue Univ., 1963 (Harper and Row, New York, 1965) 93–110.Google Scholar
27.Tate, J., ‘Conjectures on algebraic cycles in ℓ-adic cohomology’, Proc. Sympos. Pure Math. 55 (1994) Part 1, 71–83.CrossRefGoogle Scholar
28.Tsuzuki, N., ‘On the Gysin isomorphism in rigid cohomology’, Hiroshima Math. J. 29 (1999) 479–527.CrossRefGoogle Scholar
Supplementary material: File

JCM 11 Lauder Appendix 1

Lauder Appendix 1

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Supplementary material: File

JCM 11 Lauder Appendix 2

Lauder Appendix 2

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JCM 11 Lauder Appendix 3

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JCM 11 Lauder Appendix 4

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JCM 11 Lauder Appendix 5

Lauder Appendix 5

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JCM 11 Lauder Appendix 6

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