Refined Selmer equations for the thrice-punctured line in depth two

Kim gave a new proof of Siegel’s Theorem that there are only finitely many S S -integral points on P Z 1 ∖ { 0 , 1 , ∞ } \mathbb {P}^1_\mathbb {Z}\setminus \{0,1,\infty \} . One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of S S increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where S S has size 2 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.