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Reductions of an elliptic curve and their Tate-Shafarevich groups

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In this paper we study the Tate-Shafarevich groups III p of the reductions modulo primes p of an elliptic curve E/ℚ considered as being defined over their function fields. Assuming GRH when E has no CM, we show that III p is trivial for a positive proportion of primes p, provided E has an irrational point of order two.

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References

  1. Adelmann, C.: The decomposition of primes in torsion point fields. Lecture Notes in Mathematics, 1761, Springer-Verlag, Berlin, 2001

  2. Cojocaru, A. C.: Cyclicity of elliptic curves modulo p, Ph.D., 2002. Queen’s University (Kingston, Canada)

  3. Cojocaru, A. C., Fouvry, E., Ram Murty, M.: The square sieve and the Lang-Trotter conjecture. To appear in Canadian J. of Math.

  4. Cojocaru, A. C., Ram Murty, M.: Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem. Preprint

  5. Deuring, M.: Teilbarkeitseigenschaften der singulären Moduln der elliptischen Funktionen und die Diskriminante der Klassengleichung (German). Comment. Math. Helv. 19, 74–82 (1946)

    MathSciNet  MATH  Google Scholar 

  6. Duke, W., Tóth, Á.: The splitting of primes in division fields of elliptic curves. Experimental Math. 11, 555–565 (2003)

    Google Scholar 

  7. Elkies, N. D.: The existence of infinitely many supersingular primes for every elliptic curve over ℚ. Inventiones Mathematicae 89, 561–567 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Gupta, R., Ram Murty, M.: Cyclicity and generation of points modulo p on elliptic curves. Inventiones Mathematicae 101, 225–235 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Halberstam, H., Richert, H.-E.: Sieve methods, London Mathematical Society Monographs No. 4, Academic Press, London-New York, 1974

  10. Hasse, H.: Zur Geschlechtertheorie in quadratischen Zahlkörpern (German). J. Math. Soc. Japan 3, 45–51 (1951)

    MathSciNet  MATH  Google Scholar 

  11. Hooley, C.: On Artin’s conjecture. J. reine angew. Math. 225, 209–220 (1967)

    MATH  Google Scholar 

  12. Lagarias, J., Odlyzko, A.: Effective versions of the Chebotarev density theorem, in Algebraic Number Fields, A. Fröhlich (ed.) New York: Academic Press 1977, pp. 409–464

  13. Lang, S., Tate, J.: Principal homogeneous spaces over abelian varieties. Am. J. Math. 80, 659–684 (1958)

    MATH  Google Scholar 

  14. Lang, S., Trotter, H.: Frobenius distributions in GL2-extensions. Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin-New York, 1976, iii+274

  15. Milne, J. S.: The Tate-Šafarevič group of a constant abelian variety. Inventiones Mathematicae 6, 91–105 (1968)

    MATH  Google Scholar 

  16. Ram Murty, M.: On Artin’s conjecture. Journal of Number Theory 16, 147–168 (1983)

    MathSciNet  MATH  Google Scholar 

  17. Oesterlé, J.: Empilements de sphères (French) [Sphere packings] Séminaire Bourbaki, Vol. 1989/90, Astérisque No. 189-190 (1990), Exp. No. 727, pp. 375–397

  18. Roman, S.: Field theory, Graduate Texts in Mathematics 158, Springer-Verlag, New York, 1995

  19. Schoof, R.: The exponents of the groups of points on the reductions of an elliptic curve, Arithmetic algebraic geometry (Texel, 1989), 325–335, Progr. Math. 89, Birkháuser Boston, Boston, MA, 1991

  20. Serre, J-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Inventiones Mathematicae 15 (1972), 259–331, also in Collected papers, volume III, Springer-Verlag, 1985

  21. Serre, J-P.: Résumé des cours de 1977-1978, Annuaire du Collège de France 1978, 67–70, in Collected papers, volume III, Springer-Verlag, 1985

  22. Serre, J-P.: Quelques applications du théorème de densité de Chebotarev. Publ. Math. I. H. E. S., 54 (1981), 123–201, also in Collected papers, volume III, Springer-Verlag, 1985

    Google Scholar 

  23. Shioda, T.: Some remarks on elliptic curves over function fields, Journées Arithmétiques 1991 (Geneva), Astérisque No. 209, (1992), 12, 99–114

  24. Silverman, J.: The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, Berlin-New York, 1986

  25. Suzuki, M.: Group theory I, Translated from the Japanese by the author, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 247, Springer-Verlag, Berlin-New York, 1982

  26. Tate, J.: Algebraic cycles and poles of zeta functions, 1965 Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), 93–110, Harper and Row, New York

  27. Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki Vol.9, Exp. No. 306, 415–440, Soc. Math. France, Paris, 1995

  28. Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular functions of one variable IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33–52, Lecture Notes in Math. Vol. 476, Springer-Verlag, Berlin, 1975

  29. Tate, J.: The arithmetic of elliptic curves. Inventiones Mathematicae 23, 179–206 (1974)

    MATH  Google Scholar 

  30. Washington, L. C.: Introduction to cyclotomic fields, Second edition, Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997

  31. Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2(4), pp. 521–560 (1969)

    Google Scholar 

  32. Weil, A.: Sur les courbes algébriques et les variétés qui s’en déduisent (French), Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris, 1948. iv+85 pp

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Authors and Affiliations

  1. The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario, M5T 3J1, Canada

    A. C. Cojocaru

  2. Department of Mathematics, University of California, 951555, Los Angeles, CA 90095-1555, USA

    W. Duke

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  1. A. C. Cojocaru
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  2. W. Duke
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Correspondence to W. Duke.

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Research supported in part by an NSERC postdoctoral fellowship.

Research supported in part by NSF grant DMS-98-01642.

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Cojocaru, A., Duke, W. Reductions of an elliptic curve and their Tate-Shafarevich groups. Math. Ann. 329, 513–534 (2004). https://doi.org/10.1007/s00208-004-0517-2

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