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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

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.

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 -integral points on . 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 increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where has size which has been introduced in [BD19]. In so doing, we exhibit new examples of a natural generalisation of a conjecture of Kim.

Paper Structure

This paper contains 17 sections, 20 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $S$-Unit Equation and Classification of Solutions
  3. Refined Selmer Schemes and the $S_3$-Action
  4. The Chabauty--Kim method
  5. Chabauty--Kim in depth $n \leq 2$ for $\mathcal{X} = \mathbb{P}^1\setminus\{0,1,\infty\}$
  6. Depth 1
  7. Depth 2
  8. The localization map in depth 2
  9. Examples
  10. Refined Selmer Schemes
  11. The $S_3$-action
  12. Explicit Equations
  13. The case $S = \{2\}$
  14. Sets $S$ of size two
  15. Power series
  16. ...and 2 more sections

Key Result

Theorem A

Let $S=\{2\}$ and let $p$ be an odd prime. Then $\mathcal{X}(\mathbb{Z}_p)^{\min}_{S,2}$ is equal to the set of nontrivial $(p-1)$-st roots of unity $\zeta \in \mathbb{Z}_p$ for which $\mathop{\mathrm{Li}}\nolimits_2(\zeta) = 0$, along with all the images of this set under the natural action of $S_3

Figures (1)

  • Figure 1: The three components of the Zariski-closure of the image of $j_\ell$ correspond to the rays of the tropicalization.

Theorems & Definitions (45)

  • Theorem A: Proposition \ref{['prop:refinedCKset_card=1']}
  • Theorem B: Theorem \ref{['refined Chabauty-Kim equations']}
  • Proposition : Proposition \ref{['newton polygon analysis result']}
  • Remark : Remark \ref{['rem:empty_locus']}
  • Proposition 1.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: DCW
  • ...and 35 more