A proof that ℤ is Diophantine in ℚ through congruent-number twists

We give a proof of Glazer and collaborators' theorem that ℤ is Diophantine in ℚ, which builds on OpenAI's proof that Hilbert's tenth problem over ℚ has a negative answer. Integers are represented by multiples of a point on an elliptic curve of rank one over ℚ(√2). An existential condition on two congruent-number twists forbids poles of odd order at odd primes. A height bound at six rational points, from a quaternionic Shimura curve and Serre's conjecture (a theorem of Khare–Wintenberger and Kisin), and a prime pattern from the Green–Tao–Ziegler theorem turn congruences at chosen primes into identities between multiples, and the growth of a Lucas sequence forces integrality. Our contribution is the existence of the rational points that the condition on twists requires: for coprime squarefree integers β > 1 and θ ≥ 1 with β ≡ θ ≡ 1 (mod 8), each a square modulo the primes of the other, there is an integer e such that y² = x³ − e²x and y² = x³ − e²θ²x both have rank one and a rational point with abscissa in β·(ℚ×)². Where Glazer and collaborators use a restricted 2-converse theorem, this rests on published theorems: the genus criterion of Tian, Yuan and Zhang, with Gross–Zagier.