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

Abstract

In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on ${\mathbb{P}}_{\mathbb{Z}}^1\setminus \{0,1,\infty \}$. 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 $S$ increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where $S$ has size $2$ which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.

35 pages, comments welcome

Cited by in corpus (2)

• Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$
• Linear and quadratic Chabauty for affine hyperbolic curves