Inventiones mathematicae

, Volume 181, Issue 3, pp 541–575 | Cite as

Ranks of twists of elliptic curves and Hilbert’s tenth problem

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Article

Abstract

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.

References

  1. 1.
    Cassels, J.W.S.: Arithmetic on an elliptic curve. In: Proc. Internat. Congr. Mathematicians, Stockholm, 1962, pp. 234–246. Inst. Mittag-Leffler, Djursholm (1963) Google Scholar
  2. 2.
    Cassels, J.W.S.: Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math. 217, 180–199 (1965) MATHMathSciNetGoogle Scholar
  3. 3.
    Chang, S.: On the arithmetic of twists of superelliptic curves. Acta Arith. 124, 371–389 (2006) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chang, S.: Quadratic twists of elliptic curves with small Selmer rank. Preprint available at http://arxiv.org/abs/0809.5019
  5. 5.
    Denef, J.: Diophantine sets over algebraic integer rings. II. Trans. Am. Math. Soc. 257, 227–236 (1980) MATHMathSciNetGoogle Scholar
  6. 6.
    Denef, J., Lipshitz, L.: Diophantine sets over some rings of algebraic integers. J. Lond. Math. Soc. 18, 385–391 (1978) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dokchitser, T., Dokchitser, V.: Elliptic curves with all quadratic twists of positive rank. Acta Arith. 137, 193–197 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eisenträger, K.: Hilbert’s tenth problem and arithmetic geometry. Ph.D. thesis, UC Berkeley (2003) Google Scholar
  9. 9.
    Gouvêa, F., Mazur, B.: The square-free sieve and the rank of elliptic curves. J. Am. Math. Soc. 4, 1–23 (1991) MATHCrossRefGoogle Scholar
  10. 10.
    Heath-Brown, D.R.: The size of Selmer groups for the congruent number problem II. Invent. Math. 118, 331–370 (1994) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kramer, K.: Arithmetic of elliptic curves upon quadratic extension. Trans. Am. Math. Soc. 264, 121–135 (1981) MATHGoogle Scholar
  12. 12.
    Matiyasevich, Y.V.: The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR 191, 279–282 (1970) MathSciNetGoogle Scholar
  13. 13.
    Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18, 183–266 (1972) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mazur, B., Rubin, K.: Kolyvagin systems, Mem. Am. Math. Soc., vol. 799 (2004) Google Scholar
  15. 15.
    Mazur, B., Rubin, K.: Finding large Selmer rank via an arithmetic theory of local constants. Ann. Math. 166, 581–614 (2007) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124, 437–449 (1996) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Milne, J.S.: Arithmetic Duality Theorems. Perspectives in Math., vol. 1. Academic Press, San Diego (1986) MATHGoogle Scholar
  18. 18.
    Ono, K.: Nonvanishing of quadratic twists of modular L-functions and applications to elliptic curves. J. Reine Angew. Math. 533, 81–97 (2001) MATHMathSciNetGoogle Scholar
  19. 19.
    Ono, K., Skinner, C.: Non-vanishing of quadratic twists of modular L-functions. Invent. Math. 134, 651–660 (1998) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Poonen, B.: Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers. In: Algorithmic Number Theory, Sydney, 2002. Lecture Notes in Comput. Sci., vol. 2369, pp. 33–42. Springer, Berlin (2002) CrossRefGoogle Scholar
  21. 21.
    Rohrlich, D.: Galois theory, elliptic curves, and root numbers. Compos. Math. 100, 311–349 (1996) MATHMathSciNetGoogle Scholar
  22. 22.
    Rubin, K.: Euler Systems. Annals of Math. Studies, vol. 147. Princeton University Press, Princeton (2000) MATHGoogle Scholar
  23. 23.
    Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. Enseign. Math. 22, 227–260 (1976) MATHMathSciNetGoogle Scholar
  24. 24.
    Shlapentokh, A.: Hilbert’s tenth problem over number fields, a survey. In: Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry, Ghent, 1999. Contemp. Math., vol. 270, pp. 107–137. Am. Math. Soc., Providence (2000) Google Scholar
  25. 25.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986) MATHGoogle Scholar
  26. 26.
    Skorobogatov, A., Swinnerton-Dyer, P.: 2-descent on elliptic curves and rational points on certain Kummer surfaces. Adv. Math. 198, 448–483 (2005) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Swinnerton-Dyer, H.P.F.: The effect of twisting on the 2-Selmer group. Math. Proc. Cambridge Philos. Soc. 145, 513–526 (2008) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Tate, J.: Duality theorems in Galois cohomology over number fields. In: Proc. Intern. Congr. Math., Stockholm, pp. 234–241 (1962) Google Scholar
  29. 29.
    Wintner, A.: On the prime number theorem. Am. J. Math. 64, 320–326 (1942) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsUC IrvineIrvineUSA

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