The motivic anabelian geometry of local heights on abelian varieties

Betts, L. Alexander

doi:10.48550/arxiv.1706.04850

Cited by 3 publications

(4 citation statements)

References 32 publications

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“…These two local constancy results play a central role in the theory developed in [Bet19], which relates these pro-unipotent Kummer maps to the classical theory of Néron-Tate heights on abelian varieties. Moreover, by generalising some of the foundational results supporting the Chabauty-Kim method, Theorems 1.1 and 1.2 constitute preliminary steps towards setting up the Chabauty-Kim method for higher-dimensional varieties and, more interestingly, for varieties with bad reduction.…”

Section: Introductionmentioning

confidence: 95%

Local constancy of pro-unipotent Kummer maps

Betts¹

2022

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It is a theorem of Kim-Tamagawa that the Q -pro-unipotent Kummer map associated to a smooth projective curve Y over a finite extension of Qp is locally constant when = p. The present paper establishes two generalisations of this result. Firstly, we extend the Kim-Tamagawa Theorem to the case that Y is a smooth variety of any dimension. Secondly, we formulate and prove the analogue of the Kim-Tamagawa Theorem in the case = p, again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale-de Rham comparison theorem for pro-unipotent fundamental groupoids using methods of Scholze and Diao-Lan-Liu-Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids. Contents 1. Introduction 1 2. Local constancy of the Q -pro-unipotent Kummer map 5 3. The unipotent Tannakian formalism 6 4. The unipotent Riemann-Hilbert fundamental groupoid 11 5. Comparison between étale and de Rham fundamental groupoids 17 6. Horizontal de Rham local systems over polydiscs 25 7. Local constancy of the Q p -pro-unipotent Kummer map mod H 1 e 29 References 30

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Section: Introductionmentioning

confidence: 95%

Local constancy of pro-unipotent Kummer maps

Betts¹

2022

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“…We call v L the canonical valuation associated to (L, r). Betts calls the canonical valuation the Néron log-metric in [Bet17]. We have chosen different terminology to avoid confusion with the log functions discussed in the next few sections.…”

Section: Valuations and Canonical Local Heights Away From Pmentioning

confidence: 99%

“…v , the total space of L v without the zero section. See Definition 2.1; valuations are called log-metrics by Betts in [Bet17].…”

Section: Introductionmentioning

confidence: 99%

p-adic adelic metrics and Quadratic Chabauty I

Besser¹,

Müller²,

Srinivasan³

2021

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We give a new construction of p-adic heights on varieties over number fields using p-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of p-adic adelic metrics on line bundles. In particular, we describe a construction of canonical p-adic heights an abelian varieties and we show that, for Jacobians, this recovers the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle.

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“…If P is crystalline (i.e. B cris -admissible in the sense of [Bet17,(3.4.3)]), then it is also de Rham, and the fact that D commutes with tensor products on B dR -admissible representations [Fon94, §1.5] implies that D(P ) is naturally a U dR n -torsor over K. Here, D(P ) inherits a Hodge filtration from B dR and the action map D(P ) × U dR n → D(P ) is compatible with it. Similarly, the crystalline Dieudonné functor D cris and the comparison D cris (P ) × K 0 K ∼ = D(P ) (where K 0 is the maximal absolutely unramified subfield of K) equip D(P ) with a Frobenius automorphism compatible with the torsor structure.…”

Section: A1 a Cocycle Lifting Consider First The Mapmentioning

confidence: 99%

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

Best¹,

Betts²,

Kumpitsch³

et al. 2021

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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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The motivic anabelian geometry of local heights on abelian varieties

Cited by 3 publications

References 32 publications

Local constancy of pro-unipotent Kummer maps

Local constancy of pro-unipotent Kummer maps

p-adic adelic metrics and Quadratic Chabauty I

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

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