### Abstract

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers.

Original language | English (US) |
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Pages (from-to) | 1951-1959 |

Number of pages | 9 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

Eisenträger, K., & Everest, G. (2009). Descent on elliptic curves and hilbert's tenth problem.

*Proceedings of the American Mathematical Society*,*137*(6), 1951-1959. https://doi.org/10.1090/S0002-9939-08-09740-2