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

  • Bjorn Poonen
Conference paper

DOI: 10.1007/3-540-45455-1_4

Part of the Lecture Notes in Computer Science book series (LNCS, volume 2369)
Cite this paper as:
Poonen B. (2002) Using Elliptic Curves of Rank One towards the Undecidability of Hilbert’s Tenth Problem over Rings of Algebraic Integers. In: Fieker C., Kohel D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

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

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