Using Elliptic Curves of Rank One towards the Undecidability of Hilbert’s Tenth Problem over Rings of Algebraic Integers

  • Bjorn Poonen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2369)

Abstract

Let FK be number fields, and let \( \mathcal{O}_F \) and \( \mathcal{O}_K \) be their rings of integers. If there exists an elliptic curve E over F such that rk, E(F) = rk, E(K) = 1, then there exists a diophantine definition of \( \mathcal{O}_F \) over \( \mathcal{O}_K \) .

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References

  1. CZ00.
    Gunther Cornelissen and Karim Zahidi, Topology of Diophantine sets: remarks on Mazur’s conjectures, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 253–260.Google Scholar
  2. Dav53.
    Martin Davis, Arithmetical problems and recursively enumerable predicates, J. Symbolic Logic 18 (1953), 33–41.MATHCrossRefMathSciNetGoogle Scholar
  3. Den80.
    J. Denef, Diophantine sets over algebraic integer rings. II, Trans. Amer. Math. Soc. 257 (1980), no. 1, 227–236.MATHCrossRefMathSciNetGoogle Scholar
  4. DL78.
    J. Denef and L. Lipshitz, Diophantine sets over some rings of algebraic integers, J. London Math. Soc. (2) 18 (1978), no. 3, 385–391.MATHCrossRefMathSciNetGoogle Scholar
  5. DL+00.
    Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel (eds.), Hilbert’s tenth problem: relations with arithmetic and algebraic geometry, American Mathematical Society, Providence, RI, 2000, Papers from the workshop held at Ghent University, Ghent, November 2–5, 1999.MATHGoogle Scholar
  6. DPR61.
    Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425–436.CrossRefMathSciNetGoogle Scholar
  7. Eis.
    Kirsten Eisenträger, Ph. D. thesis, University of California, Berkeley, in preparation.Google Scholar
  8. KR92.
    K. H. Kim and F. W. Roush, Diophantine undecidability of C(t1,t2), J. Algebra 150 (1992), no. 1, 35–44.MATHCrossRefMathSciNetGoogle Scholar
  9. Mat70.
    Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279–282.MathSciNetGoogle Scholar
  10. Maz94.
    B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), no. 2, 353–371.MATHCrossRefMathSciNetGoogle Scholar
  11. MB.
    Laurent Moret-Bailly, paper in preparation, extending results presented in a lecture 18 June 2001 at a conference in honor of Michel Raynaud in Orsay, France.Google Scholar
  12. Phe88.
    Thanases Pheidas, Hilbert’s tenth problem for a class of rings of algebraic integers, Proc. Amer. Math. Soc. 104 (1988), no. 2, 611–620.MATHCrossRefMathSciNetGoogle Scholar
  13. Phe91.
    Thanases Pheidas, Hilbert’s tenth problem for fields of rational functions over finite fields, Invent. Math. 103 (1991), no. 1, 1–8.Google Scholar
  14. Phe00.
    Thanases Pheidas, An effort to prove that the existential theory of Q is undecidable, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 237–252.Google Scholar
  15. PZ00.
    Thanases Pheidas and Karim Zahidi, Undecidability of existential theories of rings and fields: a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 49–105.Google Scholar
  16. Ser97.
    Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, third ed., Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brownfrom notes by Michel Waldschmidt, With a foreword by Brown and Serre.Google Scholar
  17. Shl89.
    Alexandra Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number fields, Comm. Pure Appl. Math. 42 (1989), no. 7, 939–962.MATHCrossRefMathSciNetGoogle Scholar
  18. Shl92.
    Alexandra Shlapentokh, Hilbert’s tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic, Trans. Amer. Math. Soc. 333 (1992), no. 1, 275–298.MATHCrossRefMathSciNetGoogle Scholar
  19. Shl00a.
    Alexandra Shlapentokh, Hilbert’s tenth problem for algebraic function fields over infinite fields of constants of positive characteristic, Pacific J. Math. 193 (2000), no. 2, 463–500.MATHMathSciNetCrossRefGoogle Scholar
  20. Shl00b.
    Alexandra Shlapentokh, Hilbert’s tenth problem over number fields, a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. Soc., Providence, RI, 2000, pp. 107–137.Google Scholar
  21. Sil92.
    Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original.Google Scholar
  22. Vid94.
    Carlos R. Videla, Hilbert’s tenth problem for rational function fields in characteristic 2, Proc. Amer. Math. Soc. 120 (1994), no. 1, 249–253.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Bjorn Poonen
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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