The Unipotent Chabauty-Kim-Kantor Method for Relative Completions

TL;DR

This paper refines Kantor's relative completion approach, unifying it with Chabauty-Kim and Lawrence-Venkatesh methods, to provide a new p-adic analytic technique for computing rational points on hyperbolic curves, advancing Diophantine finiteness results.

Contribution

It introduces a unification of the Chabauty-Kim and Lawrence-Venkatesh methods via relative completion, overcoming previous limitations and enabling explicit computation of rational points.

Findings

Established density of Kantor's p-adic period map

Extended applicability to all curves of genus g >= 2

Resolved the representability problem for Kantor's global Selmer stack

Abstract

Kantor's Thesis was the first step in unifying the Chabauty-Kim and Lawrence-Venkatesh methods via relative completion. In this work, we refine Kantor's approach by addressing its limitations, achieving the first unification where a dimension inequality between local and global Galois cohomology implies Diophantine finiteness for curves. This results in a new conditional proof of Faltings' and Siegel's theorems and introduces a novel p-adic analytic method for computing rational points on hyperbolic curves, offering advantages over the Chabauty-Kim and Lawrence-Venkatesh approaches. Our technical contributions are threefold. First, we establish the density of Kantor's p-adic period map. Second, our method applies to all curves of genus g >= 2, extending beyond the specific modular curves considered in Kantor's thesis. Third, we resolve the representability problem for Kantor's global Selmer stack. Previously, Kantor's method required additional conjectures in p-adic Hodge theory to represent his Selmer stack--a priori a rigid analytic stack--in a category with a suitable dimension theory. We overcome this by replacing the unipotent completions used in Kim's framework with the unipotent radicals of Kantor's relative completions, derived from monodromy representations associated with the relative cohomology of a Kodaira-Parshin family. Kantor's method is a step toward the Effective Faltings Problem, which seeks not only to establish finiteness of rational points on hyperbolic curves but to compute them explicitly. While the Lawrence-Venkatesh method is unconditional, it has not yet been made effective for any curve. In contrast, the Chabauty-Kim method, though conditional, has been made effective in various settings. Our Unipotent Chabauty-Kim-Kantor method addresses key challenges in Kantor's program and highlights its potential for both theoretical and computational advances.