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

摘要

Let F in contained in K be number fields, and let O_F and 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 O_F over O_K.