The Unipotent Albanese Map and Selmer Varieties for Curves

Kim, Minhyong

doi:10.2977/prims/1234361156

Cited by 99 publications

(201 citation statements)

References 30 publications

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“…In [17], section 1, Lemma 2 and Lemma 3, we gave a description of a universal pro-unipotent bundle with connection on X Q p . Let α be an invariant differential 1-form on E and let β be a differential of the second kind with a pole only at e such that [ …”

Section: Preliminary Formulasmentioning

confidence: 99%

Massey products for elliptic curves of rank 1

Kim¹

2010

J. Amer. Math. Soc.

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The author must begin with an apology for writing on a topic so specific, so elementary, and so well-understood as the study of elliptic curves of rank 1. Nevertheless, it is hoped that a contribution not entirely without value or novelty is to be found within the theory of Selmer varieties for hyperbolic curves, applied to the complement X = E \ {e} of the origin inside an elliptic curve E over Q with Mordell-Weil rank 1. Assume throughout this paper that p is an odd prime of good reduction such that X(E)[p ∞ ] is finite and that E has integral j-invariant. All of these assumptions will hold, for example, if E has complex multiplication and ord s=1 L(E, s) = 1.Let E be a regular Z-model for E and X the complement in E of the closure of e. The main goal of the present inquiry is to find explicit analytic equations defining X (Z) inside X (Z p ). The approach of this paper makes use of a rigidified Massey product in Galois cohomology.1 That is, theétale local unipotent Albanese map2 ) to the level-two local Selmer variety (recalled below) associates to point z a nonabelian cocycle a(z), which can be broken canonically into two components a(z) = a 1 (z) + a 2 (z), with a 1 (z) taking values in to construct a function on the local points of X . Recall that Massey products are secondary cohomology products arising in connection with associative differential graded algebras (A,

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Section: Preliminary Formulasmentioning

confidence: 99%

Massey products for elliptic curves of rank 1

Kim¹

2010

J. Amer. Math. Soc.

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“…Starting with his groundbreaking works [Kim1,Kim2], Minhyong Kim has developed a new approach to the study of integral points, which seeks to make effective use of the nonabelian nature of the fundamental group to bound, and hopefully compute, the set of integral points. The hope to do so was already inherent in Grothendieck's section conjecture concerning the profinite fundamental group.…”

Section: Letmentioning

confidence: 99%

“…A certain lifting of the Bloch-Kato exponential map to the unipotent level [Kim2] places α and κ p in a triangle ( * ) X(Z p )…”

Section: Letmentioning

confidence: 99%

Mixed Tate Motives and the Unit Equation

Dan-Cohen

Wewers

2015

Int Math Res Notices

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Abstract. 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 equation whose halting will be conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via p-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm. We also work out an elaborate example, which goes beyond the cases that were understood before, and allows us to verify Kim's conjecture in a range of new cases.

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“…These have smooth, proper formal lifts to Γ + , and we may look at the universal n-unipotent (and integrable) ∇-modules on these lifts, similarly to Besser's work (see [1]). The natural expectation is that the map which we get this way is independent of the formal lift to Γ + , it is injective on residue disks, and it is possible to prove a suitable analogue of the main result of Kim's article [3,Thm. 1].…”

Section: Iterated P-adic Line Integrals Over Laurent Series Fields Ofmentioning

confidence: 99%

Iterated line integrals over Laurent series fields of characteristic p

Pál¹

2017

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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A l g è b r e e t t h é o r i e d e s n o m b r e s Ambrus Pál Iterated line integrals over Laurent series fields of characteristic ITERATED LINE INTEGRALS OVER LAURENT SERIES FIELDS OF CHARACTERISTIC P by Ambrus PálAbstract. -Inspired by Besser's work on Coleman integration, we use ∇-modules to define iterated line integrals over Laurent series fields of characteristic p taking values in double cosets of unipotent n × n matrices with coefficients in the Robba ring divided out by unipotent n × n matrices with coefficients in the bounded Robba ring on the left and by unipotent n×n matrices with coefficients in the constant field on the right. We reach our definition by looking at the analogous theory for Laurent series fields of characteristic 0 first, and reinterpreting the classical formal logarithm in terms of ∇-modules on formal schemes. To illustrate that the new p-adic theory is non-trivial, we show that it includes the p-adic formal logarithm as a special case.Résumé. -En nous inspirant du travail de Besser sur l'intégration de Coleman, nous utilisons les ∇-modules pour définir des intégrales curvilignes itérées sur des corps de séries de Laurent en caractéristique p qui prennent leurs valeurs dans des doubles classes de l'espace des matrices unipotentes de taille n × n à coefficients dans l'anneau de Robba, quotienté à gauche par l'ensemble des matrices unipotentes à coefficients dans l'anneau de Robba borné, et à droite par les matrices unipotentes à coefficients dans le corps de constantes. Nous aboutissons à cette définition en étudiant la théorie analogue pour les corps de séries de Laurent en caractéristique 0 puis en réinterprétant le logarithme formel classique en terme de ∇-modules sur les schémas formels. Pour montrer que cette nouvelle théorie p-adique n'est pas triviale, nous prouvons qu'elle contient le logarithme formel p-adique comme cas particulier.

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The Unipotent Albanese Map and Selmer Varieties for Curves

Cited by 99 publications

References 30 publications

Massey products for elliptic curves of rank 1

Massey products for elliptic curves of rank 1

Mixed Tate Motives and the Unit Equation

Iterated line integrals over Laurent series fields of characteristic p

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