Refined Chabauty–Kim computations for the thrice-punctured line over $Z[1/6]$.

If $X$ is a curve of genus at least $2$ defined over the rational numbers, we know by Faltings’s Theorem that the set $X(Q)$ of rational points is finite but we don’t know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce $p$-adic analytic functions on $X(Q_p)$ containing the rational points $X(Q)$ in their zero locus. We apply this method to solve the S-unit equation for S={2,3} and computationally verify Kim’s Conjecture for many choices of the auxiliary prime $p$.