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

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.

10 pages, Latex 2e. This article will appear in the Proceedings of the Algorithmic Number Theory Symposium V, in the Springer Lecture Notes in Computer Science series, accessible electronically at http://www.springer.de/comp/lncs/index.html

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