The Chabauty-Kim Method for Relative Completions

TL;DR

This thesis extends the Chabauty-Kim method to the relative completion of motivic fundamental groups, incorporating reductive quotients and unipotent parts, aiming to unify approaches to Diophantine problems like Mordell's conjecture.

Contribution

It develops a generalized Chabauty-Kim theory for relative completions, combining reductive and unipotent aspects, and applies it to the Legendre family where previous methods fail.

Findings

Generalizes Kim's unipotent results with reductive quotients

Shows the unipotent tower reduces Frobenius centralizer issues

Applies the theory to the Legendre family, overcoming Lawrence-Venkatesh limitations

Abstract

In this thesis we develop a Chabauty-Kim theory for the relative completion of motivic fundamental groups, including Selmer stacks and moduli spaces of admissible torsors for the relative completion of the de Rham fundamental group. On one hand, this work generalizes results of Kim (and therefore Chabauty) in the unipotent case by adding a reductive quotient of the fundamental group. From this perspective, the addition of a reductive part allows one to apply Chabauty-type methods to fundamental groups with trivial unipotent completion, such as $SL_2(\mathbb{Z})$. On the other hand, the unipotent part provides a natural extension of the recent work of Lawrence and Venkatesh. We show that their concern with the centralizer of Frobenius goes away as one moves up the unipotent tower and away from the reductive world of flag varieties and the Gauss-Manin connection. One is tempted to hope that the relative completion will provide a unified proof of Mordell's conjecture that takes advantage of the two methods. Toward this end, we apply our work to the Legendre family on the projective line minus three points, a particular example where the method of Lawrence and Venkatesh fails.