Explicit Chabauty-Kim theory for the thrice punctured line in depth two

TL;DR

This paper advances the explicit nonabelian Chabauty-Kim method for the thrice punctured line, showing that a key polynomial is quadratic and providing a way to compute its coefficients using $p$-adic functions.

Contribution

It proves that the polynomial in the $h_2$ map is quadratic and offers a method to explicitly compute its coefficients via $p$-adic logarithms and dilogarithms.

Findings

The polynomial in $h_2$ is quadratic.

Coefficients can be expressed using $p$-adic logarithms.

Provides an explicit computational procedure for the polynomial.

Abstract

Let $X= \mathbb{P}^1 \setminus \{0,1,\infty\}$, and let $S$ denote a finite set of prime numbers. In an article of 2005, Minhyong Kim gave a new proof of Siegel's theorem for $X$: the set $X(\mathbb{Z}[S^{-1}])$ of $S$-integral points of $X$ is finite. The proof relies on a `nonabelian' version of the classical Chabauty method. At its heart is a modular interpretation of unipotent $p$-adic Hodge theory, given by a tower of morphisms $h_n$ between certain $\mathbb{Q}_p$-varieties. We set out to obtain a better understanding of $h_2$. Its mysterious piece is a polynomial in $2|S|$ variables. Our main theorem states that this polynomial is quadratic, and gives a procedure for writing its coefficients in terms of $p$-adic logarithms and dilogarithms.