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

Alex J. Best, L. Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Angus W. McAndrew, Lie Qian, Elie Studnia, Yujie Xu

Research output: Working paperPreprintAcademic

36 Downloads (Pure)


In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many $S$-integral points on $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. 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 $S$ 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.
Original languageEnglish
Publication statusAccepted/In press - 26-Jul-2023


  • math.NT
  • Primary 14G05, Secondary 11G55, 11Y50

Cite this