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Linear quadratic Chabauty

Stevan Gajović, J. Steffen Müller

TL;DR

The paper introduces a streamlined linearization of the quadratic Chabauty method for computing integral points on certain even-degree hyperelliptic curves by exploiting a degree-zero divisor at infinity to force a linear $p$-adic height, expressible via Coleman integrals. It extends the construction to integral points over number fields using multiple idele-class characters and a rank-condition framework, providing a practical algorithm and precision analysis. The approach is demonstrated with explicit genus-2 and real quadratic examples, and is shown to be faster and simpler than prior methods, though it encounters obstructions in certain field and twist scenarios. Overall, the work broadens the applicability of quadratic Chabauty techniques to a wider class of curves and fields, with concrete computational procedures and demonstrated successes.

Abstract

We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the $p$-adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over $\Q$ and $\Q(\sqrt{7})$.

Linear quadratic Chabauty

TL;DR

The paper introduces a streamlined linearization of the quadratic Chabauty method for computing integral points on certain even-degree hyperelliptic curves by exploiting a degree-zero divisor at infinity to force a linear -adic height, expressible via Coleman integrals. It extends the construction to integral points over number fields using multiple idele-class characters and a rank-condition framework, providing a practical algorithm and precision analysis. The approach is demonstrated with explicit genus-2 and real quadratic examples, and is shown to be faster and simpler than prior methods, though it encounters obstructions in certain field and twist scenarios. Overall, the work broadens the applicability of quadratic Chabauty techniques to a wider class of curves and fields, with concrete computational procedures and demonstrated successes.

Abstract

We present a new quadratic Chabauty method to compute the integral points on certain even degree hyperelliptic curves. Our approach relies on a nontrivial degree zero divisor supported at the two points at infinity to restrict the -adic height to a linear function; we can then express this restriction in terms of holomorphic Coleman integrals under the standard quadratic Chabauty assumption. Then we use this linear relation to extract the integral points on the curve. We also generalize our method to integral points over number fields. Our method is significantly simpler and faster than all other existing versions of the quadratic Chabauty method. We give examples over and .
Paper Structure (22 sections, 9 theorems, 58 equations)

This paper contains 22 sections, 9 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Linear quadratic Chabauty for $\mathbb{Z}$-integral points
  3. Linear quadratic Chabauty for integral points over number fields
  4. Coleman--Gross $p$-adic heights
  5. Continuous $\mathbb{Q}_p$-valued idรจle class characters
  6. Local heights when $\chi_v$ is unramified
  7. Local heights when $\chi_v$ is ramified
  8. Local heights for linear quadratic Chabauty on hyperelliptic curves
  9. Local heights for linear Quadratic Chabauty: the ramified case
  10. Local heights for linear Quadratic Chabauty: the unramified case
  11. Linear quadratic Chabauty for $\mathbb{Z}$-integral points
  12. Linear quadratic Chabauty for $\mathbb{Z}$-integral points: theory
  13. Computing $\mathbb{Z}$-integral points using linear quadratic Chabauty
  14. Precision estimates
  15. Example: A curve of genus 2
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $X\colon y^2=f(x)$ be a hyperelliptic curve over $\mathbb{Q}$ such that $f\in \mathbb{Z}[x]$ is a squarefree polynomial of degree $2g+2$ whose leading coefficient is a square in $\mathbb{Z}$. Suppose $r=g$. Let $p>g$ be a prime of good reduction for $X$. Then there is a nontrivial computable loc

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • ...and 15 more