Let be a finite field and smooth, geometrically irreducible, proper curves over and set . We consider Brauer–Manin and abelian descent obstructions to the existence of rational points and to weak approximation for the curve . 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 is less than that of . We also show that we can identify the points corresponding to non-constant maps using Frobenius descents.
Soit un corps fini et , courbes lisses, géométriquement irréductibles, propres sur et soit . Nous considérons les obstructions de Brauer–Manin et de descente abélienne à l’existence de points rationnels et d’approximation faible pour la courbe . 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 est inférieur à celui de . On montre aussi que l’on peut identifier les points correspondant aux morphismes non constants en utilisant la descente de Frobenius.
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Keywords: Rational points, Brauer–Manin obstruction, Global fields
Mot clés : Points rationels, obstruction de Brauer-Manin, corps globaux
Creutz, Brendan 1; Voloch, José Felipe 1
@article{AIF_2022__72_1_43_0, author = {Creutz, Brendan and Voloch, Jos\'e Felipe}, 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}, number = {1}, year = {2022}, doi = {10.5802/aif.3473}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3473/} }
TY - JOUR AU - Creutz, Brendan AU - Voloch, José Felipe TI - The Brauer–Manin obstruction for constant curves over global function fields JO - Annales de l'Institut Fourier PY - 2022 SP - 43 EP - 58 VL - 72 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3473/ DO - 10.5802/aif.3473 LA - en ID - AIF_2022__72_1_43_0 ER -
%0 Journal Article %A Creutz, Brendan %A Voloch, José Felipe %T The Brauer–Manin obstruction for constant curves over global function fields %J Annales de l'Institut Fourier %D 2022 %P 43-58 %V 72 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3473/ %R 10.5802/aif.3473 %G en %F AIF_2022__72_1_43_0
Creutz, Brendan; Voloch, José Felipe. The Brauer–Manin obstruction for constant curves over global function fields. Annales de l'Institut Fourier, Volume 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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