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

@inproceedings{Best2021RefinedSE, title={Refined Selmer equations for the thrice-punctured line in depth two}, author={Alex J Best and L. Alexander Betts and Theresa Kumpitsch and Martin Ludtke and Angus McAndrew and Li Qian and Elie Studnia and Yujie Xu}, year={2021} }

In [Kim05], Kim gave a new proof of Siegel’s Theorem that there are only finitely many S-integral points on PZ \ {0, 1,∞}. 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…

## 2 Citations

### Restriction of Scalars Chabauty and the $S$-Unit Equation

- Mathematics
- 2020

Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound #$ X(K)$ when $\text{rank} J(K) <…

### Linear and quadratic Chabauty for affine hyperbolic curves

- Mathematics
- 2023

. We give suﬃcient conditions for ﬁniteness of linear and quadratic reﬁned Chabauty– Kim loci of aﬃne hyperbolic curves. We achieve this by constructing depth ≤ 2 quotients of the fundamental group,…

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