Weight filtrations on Selmer schemes and the effective Chabauty–Kim method

Betts, L. Alexander

doi:10.1112/s0010437x2300725x

Cited by 3 publications

References 51 publications

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

Best,

Betts,

Kumpitsch

et al. 2023

Math. Comp.

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Kim gave a new proof of Siegel’s Theorem that there are only finitely many S S -integral points on P Z 1 ∖ { 0 , 1 , ∞ } \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 S increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where S S has size 2 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.

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

Best,

Betts,

Kumpitsch

et al. 2023

Math. Comp.

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Linear and Quadratic Chabauty for Affine Hyperbolic Curves

Leonhardt,

Lüdtke,

Müller

2023

International Mathematics Research Notices

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We give sufficient conditions for finiteness of linear and quadratic refined Chabauty–Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of Balakrishnan–Dogra in the projective case. We also apply Betts’ machinery of weight filtrations to give unconditional explicit upper bounds on the number of $S$-integral points when our conditions are satisfied.

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

Best¹,

Betts²,

Kumpitsch³

et al. 2021

Preprint

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In [Kim05], Kim gave a new proof of Siegel's Theorem that there are only finitely many S-integral points on P 1 Z \ {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 generalisation of a conjecture of Kim. Contents 0. Introduction 1 1. The S -Unit Equation and Classification of Solutions 4 2. Refined Selmer Schemes and the S 3 -Action 6 3. Explicit Equations 23 Appendix A. Elementary Proof of Bilinearity 33 Appendix B. Functoriality of the Chabauty-Kim diagram 41 References 55 1. The S -Unit Equation and Classification of SolutionsBefore we begin the paper proper, let us recall a few elementary facts about S-integral points on the thrice-punctured line. S-integral points on X are the same thing as solutions to the S-unit equation, i.e. they are elements u ∈ Z × S such that 1 − u ∈ Z × S also. Equivalently, Sintegral points on X correspond to solutions (a, b, c) of the equationwith a, b, c ∈ Z coprime and divisible only by primes in S (up to identifying (a, b, c) ∼ (−a, −b, −c)).The solutions to the S-unit equation can be determined by elementary means when |S| ≤ 2.Assume firstly that S = {ℓ}. Then in any solution to (1.1), one of a, b, c must be ±ℓ n for some n ≥ 0, and the other two must be ±1. Thus, if ℓ is odd, there are no solutions for parity reasons, while if ℓ = 2 then the only solution, up to signed permutation of a, b, c, is 1 + 1 = 2. This says that X (Z S ) = ∅ if S = {ℓ} with ℓ odd, and is {2, −1, 1 2 } if S = {2}.

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Weight filtrations on Selmer schemes and the effective Chabauty–Kim method

Cited by 3 publications

References 51 publications

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

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

Linear and Quadratic Chabauty for Affine Hyperbolic Curves

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

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