Refined Chabauty--Kim computations for the thrice-punctured line over $\mathbb{Z}[1/6]$

Abstract

The Chabauty--Kim method and its refined variant by Betts and Dogra aim to cut out the $S$-integral points $X({\mathbb{Z}}_S)$ on a curve inside the $p$-adic points $X({\mathbb{Z}}_p)$ by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line when $S$ contains two primes. We describe an algorithm for computing refined Chabauty--Kim loci and verify Kim's conjecture over $\mathbb{Z}[1/6]$ for all choices of auxiliary prime $p<10,000$.

arXiv comment24 pages; minor revision; comments welcome

References in corpus (3)

• Explicit Chabauty-Kim theory for the thrice punctured line in depth two
• Mixed Tate motives and the unit equation II
• Refined Selmer equations for the thrice-punctured line in depth two