Algebraic families of nonzero elements of Shafarevich-Tate groups

DOI10.1090/S0894-0347-99-00315-XzbMath0951.11022arXivmath/9901038OpenAlexW1502502272MaRDI QIDQ4700179

Bjorn Poonen, Jean-Louis Colliot-Thélène

Publication date: 1 November 1999

Published in: Journal of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9901038

zbMATH Keywords

abelian varietyrational pointsHasse principlecubic surfaceShafarevich-Tate groups

Mathematics Subject Classification ID

Abelian varieties of dimension (> 1) (11G10) Jacobians, Prym varieties (14H40) Elliptic surfaces, elliptic or Calabi-Yau fibrations (14J27) [https://portal.mardi4nfdi.de/w/index.php?title=+Special%3ASearch&search=%22Curves+of+arbitrary+genus+or+genus+%28%0D%0Ae+1%29+over+global+fields%22&go=Go Curves of arbitrary genus or genus ( e 1) over global fields (11G30)] Varieties over global fields (11G35)

Related Items (19)

Exhibiting SHA[2 on hyperelliptic Jacobians] ⋮ Homogeneous spaces and degree 4 del Pezzo surfaces ⋮ Curves over every global field violating the local-global principle ⋮ Quadratic points on intersections of two quadrics ⋮ A positive proportion of locally soluble hyperelliptic curves over ℚ have no point over any odd degree extension ⋮ The Massey vanishing conjecture for number fields ⋮ There are genus one curves violating Hasse principle over every number field ⋮ Brauer-Manin obstruction for zero-cycles on certain varieties ⋮ Arithmetic of Châtelet surface bundles revisited ⋮ Period and index for higher genus curves ⋮ Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups ⋮ The arithmetic of certain del Pezzo surfaces and \(K3\) surfaces ⋮ Points having the same residue field as their image under a morphism ⋮ The arithmetic of certain quartic curves ⋮ On the fibration method for zero-cycles and rational points ⋮ Existence of rational points on smooth projective varieties ⋮ An explicit algebraic family of genus-one curves violating the Hasse principle ⋮ Étale Brauer-Manin obstruction for Weil restrictions ⋮ Applications of the fibration method to the Brauer–Manin obstruction to the existence of zero-cycles on certain varieties

Cites Work

• Indices of hyperelliptic curves over \(p\)-adic fields
• Real and étale cohomology
• Fibration method and Manin obstruction
• The Cassels-Tate pairing on polarized Abelian varieties
• Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Rédigé avec la colloboration de J. Dieudonné
• Good reduction of abelian varieties
• The irreducibility of the space of curves of a given genus
• Duality theorems for curves over p-adic fields
• Weak approximation and non-abelian fundamental groups
• Two special cubic surfaces
• Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète
• Néron Models
• Geometric Invariant Theory
• The Brauer group of cubic surfaces
• On the Hasse principle for cubic surfaces
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