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

Let $X={\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$, 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 $2\mathrm{\mid }S\mathrm{\mid }$ 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.