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Diophantine stability for curves over finite fields

Francesc Bars, Joan Carles Lario, Brikena Vruoni

TL;DR

The paper studies Diophantine stability (DS) for curves over finite fields, where a DS-curve $C$ satisfies $C(\mathbb{F}_{q^m})=C(\mathbb{F}_q)$ for some $m>1$. It develops a zeta-function framework, showing how the Frobenius/Weil polynomials $P(t)$, $L(t)$, and the real Weil polynomial $h(x)$ govern DS behavior, and proves finiteness results for DS-curves of fixed genus. It then analyzes numerous families, including low-genus curves, Deligne–Lusztig curves (Hermitian, Suzuki, Ree, Drinfeld), and $M$-torsion Carlitz/Drinfeld curves, providing explicit DS-curves and constructive methods to generate DS-curves of higher rank. The work yields complete genus-3 classifications, catalogs DS-curve examples across DL families, and offers mechanisms to produce DS-curves with arbitrarily large rank via base-change, highlighting implications for arithmetic in characteristic $p$ and the Langlands program in positive characteristic.

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.

Diophantine stability for curves over finite fields

TL;DR

The paper studies Diophantine stability (DS) for curves over finite fields, where a DS-curve satisfies for some . It develops a zeta-function framework, showing how the Frobenius/Weil polynomials , , and the real Weil polynomial govern DS behavior, and proves finiteness results for DS-curves of fixed genus. It then analyzes numerous families, including low-genus curves, Deligne–Lusztig curves (Hermitian, Suzuki, Ree, Drinfeld), and -torsion Carlitz/Drinfeld curves, providing explicit DS-curves and constructive methods to generate DS-curves of higher rank. The work yields complete genus-3 classifications, catalogs DS-curve examples across DL families, and offers mechanisms to produce DS-curves with arbitrarily large rank via base-change, highlighting implications for arithmetic in characteristic and the Langlands program in positive characteristic.

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.
Paper Structure (15 sections, 14 theorems, 86 equations, 1 figure)

This paper contains 15 sections, 14 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic properties
  3. Low genus curves
  4. Genus 1
  5. Genus 2
  6. Genus 3
  7. Deligne-Lusztig curves
  8. Hermitian curves (type ${}^2\!A_2$).
  9. Suzuki curves (type ${}^2\!B_2$)
  10. Ree curves (type ${}^2G_2$)
  11. Drinfeld curve
  12. $M$-torsion Carlitz curves
  13. $M$-torsion Drinfeld curves
  14. Examples of Drinfeld DS-curves of rank 3
  15. Examples of Drinfeld DS-curves of arbitrary large rank

Key Result

Proposition 2.1

Let $C/\mathbb F_q$ be a curve of genus $g\geq 1$ such that $C(\mathbb F_q)=C(\mathbb F_{q^m})$ for some $m>1$. Then, the pair $(q,m)$ has to be chosen from a finite set depending only on $g$.

Figures (1)

  • Figure 1: The real Weil polynomial $h(x) =x^5+6 x^4+10 x^3-8 x \,.$

Theorems & Definitions (32)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 22 more