Diophantine stability for curves over finite fields

TL;DR

This paper surveys curves over finite fields that maintain their rational points under field extensions, providing general results and analyzing specific families to understand their Diophantine stability.

Contribution

It introduces a comprehensive survey of Diophantine stable curves over finite fields, including new general results and analysis of specific curve families.

Findings

Certain families of curves are shown to be Diophantine stable.

General criteria for Diophantine stability are derived.

The stability property is preserved under specific conditions.

Abstract

We carry out a survey on curves defined over finite fields that are Diophantine stable; that is, with the property that the set of points of the curve is not altered under a proper field extension. First, we derive some general results of such curves and then we analyze several families of curves that happen to be Diophantine stable.