• Corpus ID: 235485159

# 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}
}
• Published 18 June 2021
• Mathematics
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
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 $\text{rank} J(K) < • Mathematics • 2023 . We give suﬃcient conditions for ﬁniteness of linear and quadratic reﬁned Chabauty– Kim loci of aﬃne hyperbolic curves. We achieve this by constructing depth ≤ 2 quotients of the fundamental group, ## References SHOWING 1-10 OF 33 REFERENCES • Mathematics • 2012 Let X=P1∖{0,1,∞} , and let S denote a finite set of prime numbers. In an article of 2005, Kim gave a new proof of Siegel's theorem for X : the set X(Z[S−1]) of S ‐integral points of X is finite. The We develop an effective version of the Chabauty--Kim method which gives explicit upper bounds on the number of$S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch--Kato • Mathematics • 2016 This is the second installment in a sequence of articles devoted to “explicit ChabautyKim theory” for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit • Mathematics • 2019 We study the Galois action on paths in the$\mathbb{Q}_\ell$-pro-unipotent \'etale fundamental groupoid of a hyperbolic curve$X$over a$p$-adic field with$\ell\neq p$. We prove an We study the unipotent Albanese map that associates the torsor of paths for p-adic fundamental groups to a point on a hyperbolic curve. It is shown that the map is very transcendental in nature, We establish a Tannakian formalism of p-adic multiple polylogarithms and p-adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de Rham fundamental torsor We study the problem of describing local components of height functions on abelian varieties over characteristic$0\$ local fields as functions on spaces of torsors under various realisations of a
The finite nth polylogarithm lin(z) ∈ ℤ/p(z) is defined as ∑k=1p−1zk/kn. We state and prove the following theorem. Let Lik: ℂp → ℂp be the p-adic polylogarithms defined by Coleman. Then a certain
• Mathematics
Mathematische Annalen
• 2018
Let X denote a hyperbolic curve over $$\mathbb {Q}$$Q and let p denote a prime of good reduction. The third author’s approach to integral points, introduced in Kim (Invent Math 161:629–656, 2005;
Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I,