Date: 21 Jun 2002

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

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Let FK 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.