Diophantine stability for elliptic curves on average

TL;DR

The paper investigates ℓ-diophantine stability for elliptic curves over number fields from a statistical viewpoint. By leveraging the Mazur–Rubin criterion and density results on residual Galois representations (notably surjectivity results of Duke), it shows that for every ℓ ≥ 5 and number field K, almost all elliptic curves E/ℚ become ℓ-diophantine stable over K, with rank-1 curves contributing a positive lower density. This stability, via Shlapentokh’s criterion, yields integrally diophantine extensions L/K and hence negative answers to Hilbert's Tenth Problem for rings of integers O_L in infinitely many cyclic extensions. The work thereby blends arithmetic statistics, Galois theory, and Diophantine definability to establish average-case stability phenomena and their consequences for decidability in number rings.

Abstract

Let $K$ be a number field and $\ell \geq 5$ a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety $X_{/K}$ at a prime $\ell$. We show that there is a positive density set of elliptic curves $E_{/\mathbb{Q}}$ of rank $1$ such that $E_{/K}$ is diophantine stable at $\ell$. This has implications for Hilbert's Tenth Problem over $\mathscr{O}_K$. This problem asks whether there exists an algorithm that decides in finite time whether a finite system of Diophantine equations over $\mathscr{O}_K$ has a solution.

Theorems & Definitions (31)

• Theorem 1.1: Mazur-Rubin mazur2018diophantine, Theorem 1.2
• Corollary 1.2: Mazur-Rubin mazur2018diophantine, Corollary 1.6
• Theorem A: Theorem \ref{['our main result']}
• Theorem 1.3: Bhargava-Skinner BhargavaSkinner
• Theorem B: Theorem \ref{['main thm 2.6']}
• Theorem C
• Definition 2.1
• Theorem 2.2: Mazur-Rubin
• proof
• Theorem 2.3