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Refined Selmer equations for the thrice-punctured line in depth two
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by Alex J. Best, L. Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Angus W. McAndrew, Lie Qian, Elie Studnia and Yujie Xu;

Math. Comp. 93 (2024), 1497-1527

DOI: https://doi.org/10.1090/mcom/3898

Published electronically: October 24, 2023

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Abstract:

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 by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.

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