math.NT — Number Theory

A proof that $\mathbb{Z}$ is Diophantine in $\mathbb{Q}$ through congruent-number twists

Contributed by Manvir Jaswal

We give a proof of Glazer and collaborators' theorem that $\mathbb{Z}$ is Diophantine in $\mathbb{Q}$, which builds on OpenAI's proof that Hilbert's tenth problem over $\mathbb{Q}$ has a negative answer. Integers are represented by multiples of a point on an elliptic curve of rank one over $\mathbb{Q}(\sqrt2)$. 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 $\beta>1$ and $\theta\ge1$ with $\beta\equiv\theta\equiv1\pmod8$, each a square modulo the primes of the other, there is an integer $e$ such that $y^2=x^3-e^2x$ and $y^2=x^3-e^2\theta^2x$ both have rank one and a rational point with abscissa in $\beta\mathbb{Q}^{\times2}$. 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.