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.
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note = {reprinted with corrections from the 1967 original},
publisher = {AMS Chelsea Publishing},
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venue = {{T}iruchirapalli, 1996},
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note = {published online: 07 November 2015},
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journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {13},
year = {2000},
number = {1},
pages = {83--99},
issn = {0894-0347},
mrclass = {11G05 (14K15)},
mrnumber = {1697093},
mrreviewer = {David Harari},
doi = {10.1090/S0894-0347-99-00315-X},
zblnumber = {0951.11022},
} -
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zblnumber = {0479.14006},
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} -
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques},
title = {La descente sur les variétés rationnelles. {II}},
journal = {Duke Math. J.},
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volume = {54},
year = {1987},
number = {2},
pages = {375--492},
issn = {0012-7094},
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mrclass = {11G35 (14G25)},
mrnumber = {0899402},
mrreviewer = {Daniel Coray},
doi = {10.1215/S0012-7094-87-05420-2},
zblnumber = {0659.14028},
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques and Swinnerton-Dyer, Peter},
title = {Intersections of two quadrics and {C}hâtelet surfaces. {I}},
journal = {J. reine angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik},
volume = {373},
year = {1987},
pages = {37--107},
issn = {0075-4102},
coden = {JRMAA8},
mrclass = {11G35 (11E81 14J20 14J26)},
mrnumber = {0870307},
mrreviewer = {Noriko Yui},
zblnumber = {0622.14030},
doi = {10.1515/crll.1987.373.37},
} -
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Skorobogatov, Alexei N.},
title = {Descent on fibrations over {${\bf P}^1_k$} revisited},
journal = {Math. Proc. Cambridge Philos. Soc.},
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volume = {128},
year = {2000},
number = {3},
pages = {383--393},
issn = {0305-0041},
coden = {MPCPCO},
mrclass = {11G35 (14D10 14J20)},
mrnumber = {1744112},
mrreviewer = {Juliusz Brzezi{ń}ski},
doi = {10.1017/S0305004199004077},
zblnumber = {1024.14003},
} -
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title = {Good reduction of the {B}rauer-{M}anin obstruction},
journal = {Trans. Amer. Math. Soc.},
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volume = {365},
year = {2013},
number = {2},
pages = {579--590},
issn = {0002-9947},
coden = {TAMTAM},
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mrnumber = {2995366},
mrreviewer = {J{ö}rg Jahnel},
doi = {10.1090/S0002-9947-2012-05556-5},
zblnumber = {06154640},
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zblnumber = {1198.14021},
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Swinnerton-Dyer, Peter},
title = {Hasse principle and weak approximation for pencils of {S}everi-{B}rauer and similar varieties},
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volume = {453},
year = {1994},
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Voisin, Claire},
title = {Cohomologie non ramifiée et conjecture de {H}odge entière},
journal = {Duke Math. J.},
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volume = {161},
year = {2012},
number = {5},
pages = {735--801},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {14F43 (14C30 14E05)},
mrnumber = {2904092},
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doi = {10.1215/00127094-1548389},
zblnumber = {1244.14010},
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@article{ctx11, mrkey = {3007293},
author = {Colliot-Th{é}l{è}ne, Jean-Louis and Xu, Fei},
title = {Strong approximation for the total space of certain quadratic fibrations},
journal = {Acta Arith.},
fjournal = {Acta Arithmetica},
volume = {157},
year = {2013},
number = {2},
pages = {169--199},
issn = {0065-1036},
coden = {AARIA9},
mrclass = {11G05 (14F22 14G05)},
mrnumber = {3007293},
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