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Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers

Bjorn Poonen
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Abstract

Let F and K be number fields, with F contained in K. and let O_F and O_K be their rings of integers. If there exists an elliptic curve E over F such that E(F) and E(K) have rank 1, then there exists a diophantine definition of O_F over O_K.

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Bjorn Poonen. 2002-03-29. Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers. https://doi.org/10.1007/3-540-45455-1_4

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