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 $\mathbb P^1_{\mathbb Z}\setminus\{0,1,\infty\}$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.10145
- arXiv:
- arXiv:2106.10145
- Bibcode:
- 2021arXiv210610145B
- Keywords:
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- Mathematics - Number Theory;
- Primary 14G05;
- Secondary 11G55;
- 11Y50
- E-Print:
- 35 pages, comments welcome