Diophantine stability and second order terms

TL;DR

This paper develops a Galois-theoretic framework to analyze Diophantine stability of genus 0 curves, demonstrating that the Hilbert symbol curve is almost surely stable and revealing a bias towards instability in second order asymptotics.

Contribution

It introduces a Galois-theoretic trichotomy for Diophantine stability and applies it to specific curves, providing new insights into stability probabilities and second order behavior.

Findings

Hilbert symbol curve is Diophantine stable with probability 1

Asymptotic formula shows strong bias towards instability

Galois-theoretic trichotomy governs stability behavior

Abstract

We establish a Galois-theoretic trichotomy governing Diophantine stability for genus $0$ curves. We use it to prove that the curve associated to the Hilbert symbol is Diophantine stable with probability $1$. Our asymptotic formula for the second order term exhibits strong bias towards instability.