On the fibration method for zero-cycles and rational points

Abstract

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Thélène, Sansuc, Kato and Saito in the 1980's. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.

arXiv comment54 pages; v3: minor updates, added Remark 9.12(ii), v4: improved exposition, final version

References in corpus (12)

• Cohomologie non ramifiée et conjecture de Hodge entière
• Insufficiency of the Brauer-Manin obstruction applied to etale covers
• Algebraic families of nonzero elements of Shafarevich-Tate groups
• Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque
• The Hardy--Littlewood conjecture and rational points
• Index of varieties over Henselian fields and Euler characteristic of coherent sheaves
• Higher dimensional analogues of Châtelet surfaces
• Intersections de deux quadriques et pinceaux de courbes de genre 1
• On the square-free representation function of a norm form and nilsequences
• Principes locaux-globaux pour certaines fibrations en torseurs sous un tore
• Strong approximation with Brauer-Manin obstruction for toric varieties
• Norm forms for arbitrary number fields as products of linear polynomials

Cited by in corpus (19)

• Zéro-cycles sur les espaces homogènes et problème de Galois inverse
• Hasse principle for generalised Kummer varieties
• Four-fold Massey products in Galois cohomology
• On the square-free representation function of a norm form and nilsequences
• Supersolvable descent for rational points
• Rational points on fibrations with few non-split fibres
• Schinzel Hypothesis on average and rational points
• The Brauer-Manin obstruction for zero-cycles on K3 surfaces
• Strong approximation for a family of norm varieties
• The tight approximation property
• Weak Approximation for $0$-cycles on a product of elliptic curves
• Products of varieties with many integral points
• On the fibration method for rational points
• On the descent conjecture for rational points and zero-cycles
• Polynomials represented by norm forms via the beta sieve
• Pureté de l'approximation forte sur le corps des fonctions d'une courbe algébrique complexe
• Finitude uniforme pour les cycles de codimension 2 sur les corps de nombres
• Arithmetic purity of strong approximation for homogeneous spaces
• On Birch and Swinnerton-Dyer's cubic surfaces