Affine Chabauty I

2025/11/20 by Marius Leonhardt, Leonhardt, Marius, Martin Lüdtke +1
Computer Science · Mathematics · #11S80 #14G05 (Primary) 11G30 #14G40 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2511.15949

openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

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.

Citations

• Generalised height pairings and the Albanese kernel
• Linear quadratic Chabauty
• The local theory of unipotent Kummer maps and refined Selmer schemes
• Unlikely intersections and the Chabauty-Kim method over number fields
• A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves

Related

• Restriction of Scalars Chabauty and the S-unit equation
• Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves
• Jacobian conjecture as a problem on integral points on affine curves
• Computing integral points on hyperelliptic curves using quadratic\n Chabauty
• Effective Chabauty for symmetric powers of curves