Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

TL;DR

The paper addresses Hilbert's tenth problem over rings of integers of number fields by proving rank stability under quadratic extension, i.e., for any quadratic extension $K/F$ there exists an abelian variety $A/F$ with $\mathrm{rank} A(F) = \mathrm{rank} A(K) > 0$. This result implies that $\mathbb{Z}$ has a diophantine model over $\mathcal{O}_K$ for every number field $K$, hence Hilbert's tenth problem is undecidable over $\mathcal{O}_K$. The method uses $(1-\zeta)$-Selmer groups of Jacobians $J_n$ of curves $C_n: y^2 = x^\ell + n$ together with silent primes to control local conditions, plus a $\Sigma$-unit equation to produce $F$-rational non-torsion points on twists, constructing a positive rank over $K$. The approach builds on and extends prior work on rank stability and undecidability in rings of integers, with connections to unconditional results for specific fields and related independent approaches.

Abstract

We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.

Theorems & Definitions (2)

• Theorem 1.1
• Corollary 1.2