The Brauer–Manin obstruction for constant curves over global function fields

Creutz, Brendan; Voloch, José Felipe

doi:10.5802/aif.3473

The Brauer–Manin obstruction for constant curves over global function fields
[L’obstruction de Brauer-Manin pour des courbes constantes sur des corps de fonctions globaux]

Creutz, Brendan ¹ ; Voloch, José Felipe ¹

¹ School of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch 8140 (New Zealand)

Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 43-58.

Résumé
Abstract

Soit $𝔽$ un corps fini et $C$ , $D$ courbes lisses, géométriquement irréductibles, propres sur $𝔽$ et soit $K = 𝔽 (D)$ . Nous considérons les obstructions de Brauer–Manin et de descente abélienne à l’existence de points rationnels et d’approximation faible pour la courbe $C \otimes_{𝔽} K$ . En particulier, nous montrons que l’obstruction de Brauer–Manin est la seule obstruction à l’approximation faible et le principe de Hasse dans le cas où le genre de $D$ est inférieur à celui de $C$ . On montre aussi que l’on peut identifier les points correspondant aux morphismes non constants $D \to C$ en utilisant la descente de Frobenius.

Let $𝔽$ be a finite field and $C, D$ smooth, geometrically irreducible, proper curves over $𝔽$ and set $K = 𝔽 (D)$ . We consider Brauer–Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve $C \otimes_{𝔽} K$ . In particular, we show that Brauer–Manin is the only obstruction to weak approximation and the Hasse principle in the case that the genus of $D$ is less than that of $C$ . We also show that we can identify the points corresponding to non-constant maps $D \to C$ using Frobenius descents.

Reçu le : 2020-09-02
Révisé le : 2020-11-23
Accepté le : 2021-01-07
Publié le : 2022-07-01

DOI : 10.5802/aif.3473

Classification : 11G20, 11G30, 14H25
Keywords: Rational points, Brauer–Manin obstruction, Global fields
Mot clés : Points rationels, obstruction de Brauer-Manin, corps globaux

Affiliations des auteurs :

Creutz, Brendan ¹ ; Voloch, José Felipe ¹

¹ School of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch 8140 (New Zealand)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The {Brauer{\textendash}Manin} obstruction for constant curves over global function fields},
     journal = {Annales de l'Institut Fourier},
     pages = {43--58},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
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     doi = {10.5802/aif.3473},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3473/}
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Creutz, Brendan; Voloch, José Felipe. The Brauer–Manin obstruction for constant curves over global function fields. Annales de l'Institut Fourier, Tome 72 (2022) no. 1, pp. 43-58. doi : 10.5802/aif.3473. https://aif.centre-mersenne.org/articles/10.5802/aif.3473/

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