Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Affine Chabauty II

Marius Leonhardt, Martin Lüdtke

TL;DR

This work extends the Affine Chabauty method to compute $S$-integral points on smooth affine curves by constructing explicit logarithmic differentials whose $p$-adic integrals take prescribed values on $S$-integral points. The key mechanism is a $p$-adic residue framework that relates integrals of log differentials to residues at cusps, enabling the formation of an explicit matrix $M(U)$ whose kernel yields annihilating differentials $\omega$ and constants $c$ that bound the $S$-integral locus to a finite set of zeros in residue discs. The paper provides a detailed algorithmic roadmap (including regularizing models, reduction types, Mordell–Weil data, and vertical corrections) and demonstrates the method on concrete hyperelliptic and superelliptic families, highlighting practical computation via existing software. The results give finite, computable Chabauty loci for $S$-integral points and supply reproducible code for applying the approach to new curves, with potential impact on explicit Diophantine problem solving in arithmetic geometry.

Abstract

We present an algorithm for determining the set of $S$-integral points on an affine curve based on the Affine Chabauty method developed in the first part of this series. We achieve this by constructing explicit logarithmic differentials whose integrals take on prescribed values on $S$-integral points. Along the way, we prove a $p$-adic residue theorem for Coleman integrals of log differentials.

Affine Chabauty II

TL;DR

This work extends the Affine Chabauty method to compute -integral points on smooth affine curves by constructing explicit logarithmic differentials whose -adic integrals take prescribed values on -integral points. The key mechanism is a -adic residue framework that relates integrals of log differentials to residues at cusps, enabling the formation of an explicit matrix whose kernel yields annihilating differentials and constants that bound the -integral locus to a finite set of zeros in residue discs. The paper provides a detailed algorithmic roadmap (including regularizing models, reduction types, Mordell–Weil data, and vertical corrections) and demonstrates the method on concrete hyperelliptic and superelliptic families, highlighting practical computation via existing software. The results give finite, computable Chabauty loci for -integral points and supply reproducible code for applying the approach to new curves, with potential impact on explicit Diophantine problem solving in arithmetic geometry.

Abstract

We present an algorithm for determining the set of -integral points on an affine curve based on the Affine Chabauty method developed in the first part of this series. We achieve this by constructing explicit logarithmic differentials whose integrals take on prescribed values on -integral points. Along the way, we prove a -adic residue theorem for Coleman integrals of log differentials.
Paper Structure (14 sections, 6 theorems, 66 equations)

This paper contains 14 sections, 6 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Affine Chabauty algorithm
  4. Structure of the paper
  5. The Affine Chabauty method
  6. Strategy
  7. Intersection numbers
  8. Reduction types
  9. Selmer sets
  10. A residue theorem for $p$-adic integrals
  11. Annihilating log differentials
  12. Examples
  13. Even hyperelliptic curves
  14. A class of superelliptic curves

Key Result

Theorem 1

Let $\Sigma$ be an $S$-integral reduction type with cuspidal part $\Sigma^{{\mathrm{csp}}}$ and let $\mathfrak p \not\in S$ be a prime of $K$. If $(a_1,\ldots,a_{g+n-1}) \in K_{\mathfrak p}^{g+n-1}$ is contained in the kernel of the matrix $M(\Sigma^{{\mathrm{csp}}})$ defined in eq:MSigma--eq:matrix

Theorems & Definitions (17)

  • Theorem 1
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: $p$-adic residue theorem, case of rational cusps
  • proof
  • Remark 3.4
  • Theorem 3.5: $p$-adic residue theorem, general version
  • proof
  • Remark 4.1
  • Theorem 4.2
  • ...and 7 more