Title: The Unipotent Chabauty–Kim–Kantor Method for Relative Completions

Abstract: The Effective Siegel-Faltings Problem aims to explicitly construct, in finite computation, a complete list of S-integral points on a given hyperbolic curve. Recent advances on this problem include the groundbreaking methods of Chabauty–Kim and Lawrence–Venkatesh. While both methods essentially analyze the variation of mixed Hodge structures over natural bundles attached to the curve, these methods have complementary strengths: the Chabauty–Kim method, while conditional on Bloch–Kato, has successfully facilitated effective computation of integral points on various curves. Conversely, the Lawrence–Venkatesh method is unconditional but has not yet been practically applied to compute integral points for any specific curve. Kantor's thesis was a promising initial effort toward bridging these two methods, aiming to combine their strengths through the theory of relative completions. In joint work with David Corwin, titled "The Unipotent Chabauty–Kim–Kantor Method for Relative Completions," we present the first genuine synthesis of these powerful methods.

In this talk, we will review the Chabauty-Kim/Lawrence-Venkatesh methods, outline Kantor's initial approach, and then discuss our variant method, which resolves several key limitations identified in Kantor's work. Our main result reduces the problem of Diophantine finiteness to a dimension inequality involving a pair of algebraic spaces. If time permits, we will derive this dimension inequality explicitly for modular curves under Bloch–Kato.

Panopto link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=8e530ee3-b957-4275-8251-b39e0068d5c9