Refined Chabauty--Kim calculations for the thrice-punctured line over Z[1/6]

Martin Lüdtke*

*Corresponding author for this work

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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 $p<10{,}000$.
Original languageEnglish
PublisherarXiv
Number of pages23
Publication statusSubmitted - 5-Feb-2024

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