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

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

Let F ⊆ K 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 \( \mathcal{O}_K \) .

This research was supported by NSF grant DMS-9801104, and a Packard Fellowship.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Bjorn Poonen

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  1. Bjorn Poonen
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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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. https://doi.org/10.1007/3-540-45455-1_4

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  • DOI: https://doi.org/10.1007/3-540-45455-1_4

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