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Counting rational points on elliptic and hyperelliptic curves over function fields

Jean Gillibert, Emmanuel Hallouin, Aaron Levin

TL;DR

The paper develops explicit, characteristic-free bounds on the number of rational points of bounded height and $S$-integral points on hyperelliptic curves over function fields $k(B)$, by combining $2$-descent with Riemann–Roch and Bézout arguments. The bounds depend on the height $h_B(f)$ of the defining polynomial, the genus $g$, the irreducible components of $f$, and the Lang–Néron group $\mathrm{LN}(J)$ of the Jacobian, with refinements when the base is $\mathbb{P}^1$. A key technical ingredient is translating torsion questions for Jacobians into counts of integral points on associated curves over $k(t)$ and applying Maroni-type refinements for trigonal curves to obtain sharper results. The authors apply these bounds to bound $3$-torsion of hyperelliptic Jacobians and $2$-torsion of trigonal Jacobians over small finite fields, yielding improvements over trivial and Weil-based bounds in several regimes. Overall, the work provides explicit, effective tools to control rational/integral points and torsion in function-field settings with direct arithmetic-geometry applications.

Abstract

Combining $2$-descent techniques with Riemann-Roch and Bézout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We deduce an upper bound on the number of $S$-integral points, where $S$ is a finite set of places. As a primary application, over small finite fields we bound the $3$-torsion of Jacobians of hyperelliptic curves and the $2$-torsion of Jacobians of trigonal curves. In this setting, these bounds improve on both the trivial geometric bound and the naive inequality coming from the Weil bound, as well as recent upper bounds on $2$-torsion in the work of Bhargava et al.

Counting rational points on elliptic and hyperelliptic curves over function fields

TL;DR

The paper develops explicit, characteristic-free bounds on the number of rational points of bounded height and -integral points on hyperelliptic curves over function fields , by combining -descent with Riemann–Roch and Bézout arguments. The bounds depend on the height of the defining polynomial, the genus , the irreducible components of , and the Lang–Néron group of the Jacobian, with refinements when the base is . A key technical ingredient is translating torsion questions for Jacobians into counts of integral points on associated curves over and applying Maroni-type refinements for trigonal curves to obtain sharper results. The authors apply these bounds to bound -torsion of hyperelliptic Jacobians and -torsion of trigonal Jacobians over small finite fields, yielding improvements over trivial and Weil-based bounds in several regimes. Overall, the work provides explicit, effective tools to control rational/integral points and torsion in function-field settings with direct arithmetic-geometry applications.

Abstract

Combining -descent techniques with Riemann-Roch and Bézout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic . We deduce an upper bound on the number of -integral points, where is a finite set of places. As a primary application, over small finite fields we bound the -torsion of Jacobians of hyperelliptic curves and the -torsion of Jacobians of trigonal curves. In this setting, these bounds improve on both the trivial geometric bound and the naive inequality coming from the Weil bound, as well as recent upper bounds on -torsion in the work of Bhargava et al.
Paper Structure (13 sections, 22 theorems, 155 equations)

This paper contains 13 sections, 22 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Counting rational points
  3. Heights
  4. The $2$-descent map
  5. Upper bound for rational points of given height
  6. Proof of Theorem \ref{['thm:intro1']}
  7. Counting integral points
  8. Upper bound on the height of $S$-integral points
  9. Proof of Theorem \ref{['thm:intro2']}
  10. Integral points of small height on elliptic curves over $\mathbb{P}^1$
  11. Applications to bounding torsion of Jacobians over small finite fields
  12. Bounding $3$-torsion of Jacobians of hyperelliptic curves over small finite fields
  13. Bounding $2$-torsion of Jacobians of trigonal curves over small finite fields

Key Result

Theorem 1.1

Let $\mathcal{C}$ be the hyperelliptic curve defined over $k(B)$ by the equation $y^2=f(x)$ where $f$ is monic and separable, of odd degree $d\geq 3$. Let $h_B(f)$ be the height of $f$ (see §sec:heights). Let $C_f$ be the smooth projective curve defined over $k$ by the equation $f(x)=0$, and let $\o be the Lang-Néron group of $J$ relative to $k(B)/k$ (which, according to the Lang-Néron theorem, is

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1
  • proof
  • Proposition 2.2: Northcott
  • ...and 41 more