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Affine Chabauty I

Marius Leonhardt, Martin Lüdtke

TL;DR

Affine Chabauty I develops an S-integral analogue of Chabauty–Coleman by embedding an affine hyperbolic curve $Y$ into its generalised Jacobian $J_Y$ and constraining the Abel–Jacobi image via a local/global $D$-intersection map. By combining arithmetic intersection theory with $p$-adic logarithmic integrations of differentials, the authors produce nonzero Chabauty functions that vanish on $S$-integral points of fixed reduction type, yielding finiteness and explicit bounds under a rank–genus inequality. The framework unifies local reduction analysis (via $D$-transversal models and Selmer-type sets) with global Selmer data, and it specializes to effective bounds for even-degree hyperelliptic curves, recovering and extending the Linear Quadratic Chabauty results. The approach has computational promise: a follow-up article will describe an algorithm to compute the log differentials and their zeros, enabling determination of the $S$-integral points in practice and informing potential affine quadratic Chabauty extensions. The work thus lays foundational theory and concrete bounds for algorithmic determination of integral points on affine curves.

Abstract

We prove finiteness and give an explicit upper bound on the number of $S$-integral points on affine curves satisfying a certain rank-genus inequality. We achieve this by developing an analogue of the Chabauty method, embedding the curve into its generalised Jacobian and bounding the Abel-Jacobi image of the $S$-integral points using arithmetic intersection theory. Our results also provide the foundations for a computational method to determine the set of $S$-integral points on affine curves which will be presented in a follow-up article.

Affine Chabauty I

TL;DR

Affine Chabauty I develops an S-integral analogue of Chabauty–Coleman by embedding an affine hyperbolic curve into its generalised Jacobian and constraining the Abel–Jacobi image via a local/global -intersection map. By combining arithmetic intersection theory with -adic logarithmic integrations of differentials, the authors produce nonzero Chabauty functions that vanish on -integral points of fixed reduction type, yielding finiteness and explicit bounds under a rank–genus inequality. The framework unifies local reduction analysis (via -transversal models and Selmer-type sets) with global Selmer data, and it specializes to effective bounds for even-degree hyperelliptic curves, recovering and extending the Linear Quadratic Chabauty results. The approach has computational promise: a follow-up article will describe an algorithm to compute the log differentials and their zeros, enabling determination of the -integral points in practice and informing potential affine quadratic Chabauty extensions. The work thus lays foundational theory and concrete bounds for algorithmic determination of integral points on affine curves.

Abstract

We prove finiteness and give an explicit upper bound on the number of -integral points on affine curves satisfying a certain rank-genus inequality. We achieve this by developing an analogue of the Chabauty method, embedding the curve into its generalised Jacobian and bounding the Abel-Jacobi image of the -integral points using arithmetic intersection theory. Our results also provide the foundations for a computational method to determine the set of -integral points on affine curves which will be presented in a follow-up article.

Paper Structure

This paper contains 36 sections, 37 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Strategy
  4. Structure of the paper
  5. Local theory: The $D$-intersection map
  6. The generalised Jacobian
  7. Intersection numbers
  8. Intersection numbers
  9. The $D$-intersection map
  10. Computing Φ(F)
  11. Integral reduction types
  12. $D$-transversal models
  13. Rational reduction types
  14. Comparison with the Kummer map
  15. The case $\ell\neq p$
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\Sigma$ be an $S$-integral reduction type and let $C(\Sigma)\subseteq S$ denote the set of those $\ell \in S$ where $\Sigma_{\ell}$ is cuspidal, see def:C-Sigma. Let $p \not\in S$ be a prime. Assume Then there exist a non-zero log differential $0 \neq \omega \in \mathrm H^0(X_{\mathbb Q_p}, \Omega^1(D))$ and a constant $c \in \mathbb Q_p$ such that the function $\rho\colon \mathcal{Y}(\mathb

Theorems & Definitions (110)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 100 more