Refined Selmer equations for the thricepunctured 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:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.10145
 Bibcode:
 2021arXiv210610145B
 Keywords:

 Mathematics  Number Theory;
 Primary 14G05;
 Secondary 11G55;
 11Y50
 EPrint:
 58 pages, comments welcome