Diophantine stability for elliptic curves on average

2023/04/19 by Anwesh Ray, Ray, Anwesh, Tom Weston +1
Computer Science · Mathematics · #11G05 #11R32 #11R45 #11U05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2304.09742

openalex publication_date 2023/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

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

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