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Failure of the Hasse principle for Châtelet surfaces in characteristic $2$

Bianca Viray (2012)

Journal de Théorie des Nombres de Bordeaux

Given any global field $k$ of characteristic $2$ , we construct a Châtelet surface over $k$ that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic $2$ , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.

Faltings modular height and self-intersection of dualizing sheaf.

Atsushi Moriwaki (1995)

Mathematische Zeitschrift

Fonction de Seshadri arithmétique en géométrie d’Arakelov

Huayi Chen (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

To any adelic invertible sheaf on a projective arithmetic variety and any regular algebraic point of the arithmetic variety, we associate a function defined on $ℝ$ which measures the separation of jets on this algebraic point by the “small” sections of the adelic invertible sheaf. This function will be used to study the arithmetic local positivity.

Fonctions thêta et points de torsion des variétés abéliennes

Sinnou David (1991)

Compositio Mathematica

Formes linéaires de logarithmes effectives sur les variétés abéliennes

Éric Gaudron (2006)

Annales scientifiques de l'École Normale Supérieure