The Brauer–Manin obstruction for constant curves over global function fields
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 43-58.

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 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 DC using Frobenius descents.

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 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 DC en utilisant la descente de Frobenius.

Published online:
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
Creutz, Brendan 1; Voloch, José Felipe 1

1 School of Mathematics and Statistics University of Canterbury Private Bag 4800 Christchurch 8140 (New Zealand)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Volume 72 (2022) no. 1, pp. 43-58. doi : 10.5802/aif.3473.

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