Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers
Bjorn Poonen
Abstract
Let F and K be number fields, with F contained in K. and let OF and OK 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 OF over OK.
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