Norm forms for arbitrary number fields   as products of linear polynomials

Browning, Tim D.; Matthiesen, Lilian

doi:10.24033/asens.2348

Norm forms for arbitrary number fields as products of linear polynomials
[Représentations de normes de corps de nombres arbitraires par des produits de polynômes linéaires]

Browning, Tim D. ; Matthiesen, Lilian

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1383-1446.

Résumé
Abstract

Étant donnés un corps de nombres $K / ℚ$ et un polynôme $P \in ℚ [t]$ , dont toutes les racines sont dans $ℚ$ , soit $X$ la variété définie par l'équation $𝐍_{K} (𝐱) = P (t)$ . En combinant la combinatoire additive avec la descente, nous montrons que l'obstruction Brauer–Manin est le seul obstacle au principe de Hasse et à l'approximation faible sur un modèle projectif et lisse de $X$ .

Given a number field $K / ℚ$ and a polynomial $P \in ℚ [t]$ , all of whose roots are in $ℚ$ , let $X$ be the variety defined by the equation $𝐍_{K} (𝐱) = P (t)$ . Combining additive combinatorics with descent we show that the Brauer–Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of $X$ .

Publié le : 2017-11-18

MR Zbl

DOI : 10.24033/asens.2348

Classification : 14G05 (11B30, 11D57, 11N37, 14D10)
Keywords: additive combinatorics, Brauer–Manin obstruction, descent, Hasse principle, norm forms, weak approximation
Mot clés : Combinatoire additive, obstruction Brauer–Manin, descente, principe de Hasse, représentations de normes, approximation faible.

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     author = {Browning, Tim D. and Matthiesen, Lilian},
     title = {Norm forms for arbitrary number fields   as products of linear polynomials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1383--1446},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Browning, Tim D.; Matthiesen, Lilian. Norm forms for arbitrary number fields   as products of linear polynomials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 6, pp. 1383-1446. doi : 10.24033/asens.2348. http://www.numdam.org/articles/10.24033/asens.2348/

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