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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@book{artintate, mrkey = {2467155},
author = {Artin, Emil and Tate, John},
title = {Class Field Theory},
note = {reprinted with corrections from the 1967 original},
publisher = {AMS Chelsea Publishing},
address = {Providence, RI},
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[ctbordeaux]
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author = {Colliot-Th{é}l{è}ne, Jean-Louis},
title = {L'arithmétique du groupe de {C}how des zéro-cycles},
note = {Les Dix-huiti{è}mes Journ{é}es Arithm{é}tiques (Bordeaux, 1993)},
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mrreviewer = {Stefan M{ü}ller-Stach},
doi = {10.5802/jtnb.130},
zblnumber = {0870.14002},
} -
[cttiruchirapalli]
J. Colliot-Thélène, "The Hasse principle in a pencil of algebraic varieties," in Number Theory, Providence, RI: Amer. Math. Soc., 1998, vol. 210, pp. 19-39.
@incollection{cttiruchirapalli, mrkey = {1478483},
author = {Colliot-Th{é}l{è}ne, Jean-Louis},
title = {The {H}asse principle in a pencil of algebraic varieties},
booktitle = {Number Theory},
venue = {{T}iruchirapalli, 1996},
series = {Contemp. Math.},
volume = {210},
pages = {19--39},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {1998},
mrclass = {11G35 (14G25)},
mrnumber = {1478483},
mrreviewer = {David Harari},
doi = {10.1090/conm/210/02782},
zblnumber = {0910.14008},
} -
[ctreglees]
J. Colliot-Thélène, "Principe local-global pour les zéro-cycles sur les surfaces réglées," J. Amer. Math. Soc., vol. 13, iss. 1, pp. 101-127, 2000.
@article{ctreglees, mrkey = {1697092},
author = {Colliot-Th{é}l{è}ne, Jean-Louis},
title = {Principe local-global pour les zéro-cycles sur les surfaces réglées},
note = {with an appendix by E. Frossard and V. Suresh},
journal = {J. Amer. Math. Soc.},
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volume = {13},
year = {2000},
number = {1},
pages = {101--127},
issn = {0894-0347},
mrclass = {14C25 (11G35 11R52 14C15 14F22 14J26)},
mrnumber = {1697092},
mrreviewer = {Tam{á}s Szamuely},
doi = {10.1090/S0894-0347-99-00318-5},
zblnumber = {0972.14001},
} -
[ctbudapest]
J. Colliot-Thélène, "Points rationnels sur les fibrations," in Higher Dimensional Varieties and Rational Points, New York: Springer-Verlag, 2003, vol. 12, pp. 171-221.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis},
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booktitle = {Higher Dimensional Varieties and Rational Points},
venue = {{B}udapest, 2001},
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doi = {10.1007/978-3-662-05123-8_7},
address = {New York},
zblnumber = {1077.14029},
} -
[ctsolides]
J. Colliot-Thélène, "Zéro-cycles de degré 1 sur les solides de Poonen," Bull. Soc. Math. France, vol. 138, iss. 2, pp. 249-257, 2010.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis},
title = {Zéro-cycles de degré 1 sur les solides de {P}oonen},
journal = {Bull. Soc. Math. France},
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year = {2010},
number = {2},
pages = {249--257},
issn = {0037-9484},
coden = {BSMFAA},
mrclass = {14C25 (11G35 14F22 14G05 14G25)},
mrnumber = {2679040},
zblnumber = {1205.11075},
url = {http://smf4.emath.fr/Publications/Bulletin/138/html/smf_bull_138_249-257.php},
} -
[cthasko] J. Colliot-Thélène, D. Harari, and A. N. Skorobogatov, "Valeurs d’un polynôme à une variable représentées par une norme," in Number Theory and Algebraic Geometry, Cambridge: Cambridge Univ. Press, 2003, vol. 303, pp. 69-89.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Harari, D. and Skorobogatov, A. N.},
title = {Valeurs d'un polynôme à une variable représentées par une norme},
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series = {London Math. Soc. Lecture Note Ser.},
volume = {303},
publisher = {Cambridge Univ. Press},
address = {Cambridge},
year = {2003},
pages = {69--89},
zblnumber = {1087.14016},
mrnumber = {2053456},
} -
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Ischebeck, Friedrich},
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journal = {C. R. Acad. Sci. Paris Sér. I Math.},
fjournal = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
volume = {292},
year = {1981},
number = {15},
pages = {723--725},
issn = {0151-0509},
coden = {CRSMDY},
mrclass = {14G30 (14C15 14C25)},
mrnumber = {0618896},
mrreviewer = {Knud Lønsted},
zblnumber = {0472.14016},
} -
[cps]
J. Colliot-Thélène, A. P. Pál, and A. N. Skorobogatov, "Pathologies of the Brauer–Manin obstruction," Math. Z., 2015.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and P{á}l, A. P. and Skorobogatov, A. N.},
title = {Pathologies of the {B}rauer--{M}anin obstruction},
journal={Math. Z.},
year = {2015},
note = {published online: 07 November 2015},
doi = {10.1007/s00209-015-1565-x},
} -
[ctpoonen]
J. Colliot-Thélène and B. Poonen, "Algebraic families of nonzero elements of Shafarevich-Tate groups," J. Amer. Math. Soc., vol. 13, iss. 1, pp. 83-99, 2000.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Poonen, Bjorn},
title = {Algebraic families of nonzero elements of {S}hafarevich-{T}ate groups},
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},
} -
[ctsansuc]
J. Colliot-Thélène and J. Sansuc, "On the Chow groups of certain rational surfaces: a sequel to a paper of S. Bloch," Duke Math. J., vol. 48, iss. 2, pp. 421-447, 1981.
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques},
title = {On the {C}how groups of certain rational surfaces: a sequel to a paper of {S}. {B}loch},
journal = {Duke Math. J.},
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volume = {48},
year = {1981},
number = {2},
pages = {421--447},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {14C25 (14G05)},
mrnumber = {0620258},
mrreviewer = {Daniel Coray},
doi = {10.1215/S0012-7094-81-04824-9},
zblnumber = {0479.14006},
} -
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author = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques},
title = {Sur le principe de {H}asse et l'approximation faible, et sur une hypothèse de {S}chinzel},
journal = {Acta Arith.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
volume = {41},
year = {1982},
number = {1},
pages = {33--53},
issn = {0065-1036},
coden = {AARIA9},
mrclass = {10C02 (10B40 12B20 14G25)},
mrnumber = {0667708},
mrreviewer = {D. J. Lewis},
zblnumber = {0414.10009},
} -
[ctsandescent2]
J. Colliot-Thélène and J. Sansuc, "La descente sur les variétés rationnelles. II," Duke Math. J., vol. 54, iss. 2, pp. 375-492, 1987.
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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},
coden = {DUMJAO},
mrclass = {11G35 (14G25)},
mrnumber = {0899402},
mrreviewer = {Daniel Coray},
doi = {10.1215/S0012-7094-87-05420-2},
zblnumber = {0659.14028},
} -
[cssI]
J. Colliot-Thélène, J. Sansuc, and P. Swinnerton-Dyer, "Intersections of two quadrics and Châtelet surfaces. I," J. reine angew. Math., vol. 373, pp. 37-107, 1987.
@article{cssI, mrkey = {0870307},
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},
} -
[ctskodescent]
J. Colliot-Thélène and A. N. Skorobogatov, "Descent on fibrations over ${\bf P}^1_k$ revisited," Math. Proc. Cambridge Philos. Soc., vol. 128, iss. 3, pp. 383-393, 2000.
@article{ctskodescent, mrkey = {1744112},
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.},
fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
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},
} -
[ctskogoodreduction]
J. Colliot-Thélène and A. N. Skorobogatov, "Good reduction of the Brauer-Manin obstruction," Trans. Amer. Math. Soc., vol. 365, iss. 2, pp. 579-590, 2013.
@article{ctskogoodreduction, mrkey = {2995366},
author = {Colliot-Th{é}l{è}ne, Jean-Louis and Skorobogatov, Alexei N.},
title = {Good reduction of the {B}rauer-{M}anin obstruction},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {365},
year = {2013},
number = {2},
pages = {579--590},
issn = {0002-9947},
coden = {TAMTAM},
mrclass = {14G05 (11G25 11G35 14F22 14G25)},
mrnumber = {2995366},
mrreviewer = {J{ö}rg Jahnel},
doi = {10.1090/S0002-9947-2012-05556-5},
zblnumber = {06154640},
} -
[ctsksd98]
J. Colliot-Thélène, A. N. Skorobogatov, and P. Swinnerton-Dyer, "Rational points and zero-cycles on fibred varieties: Schinzel’s hypothesis and Salberger’s device," J. reine angew. Math., vol. 495, pp. 1-28, 1998.
@article{ctsksd98, mrkey = {1603908},
author = {Colliot-Th{é}l{è}ne, Jean-Louis and Skorobogatov, Alexei N. and Swinnerton-Dyer, Peter},
title = {Rational points and zero-cycles on fibred varieties: {S}chinzel's hypothesis and {S}alberger's device},
journal = {J. reine angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik},
volume = {495},
year = {1998},
pages = {1--28},
issn = {0075-4102},
coden = {JRMAA8},
mrclass = {14G25 (11G35 14C25 14D10)},
mrnumber = {1603908},
mrreviewer = {R. T. Hoobler},
doi = {10.1515/crll.1998.019},
zblnumber = {1198.14021},
} -
[ctsd94]
J. Colliot-Thélène and P. Swinnerton-Dyer, "Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties," J. reine angew. Math., vol. 453, pp. 49-112, 1994.
@article{ctsd94, mrkey = {1285781},
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},
journal = {J. reine angew. Math.},
fjournal = {Journal für die Reine und Angewandte Mathematik},
volume = {453},
year = {1994},
pages = {49--112},
issn = {0075-4102},
coden = {JRMAA8},
mrclass = {11G35 (11G45 14C25 14G05 14G25)},
mrnumber = {1285781},
mrreviewer = {Wayne Raskind},
doi = {10.1515/crll.1994.453.49},
zblnumber = {0805.14010},
} -
[ctvoisin]
J. Colliot-Thélène and C. Voisin, "Cohomologie non ramifiée et conjecture de Hodge entière," Duke Math. J., vol. 161, iss. 5, pp. 735-801, 2012.
@article{ctvoisin, mrkey = {2904092},
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.},
fjournal = {Duke Mathematical Journal},
volume = {161},
year = {2012},
number = {5},
pages = {735--801},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {14F43 (14C30 14E05)},
mrnumber = {2904092},
mrreviewer = {Christian Schnell},
doi = {10.1215/00127094-1548389},
zblnumber = {1244.14010},
} -
[ctx11]
J. Colliot-Thélène and F. Xu, "Strong approximation for the total space of certain quadratic fibrations," Acta Arith., vol. 157, iss. 2, pp. 169-199, 2013.
@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},
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author = {Wei, Dasheng},
title = {On the equation {$N\sb {K/k}(\Xi)=P(t)$}},
journal = {Proc. Lond. Math. Soc.},
fjournal = {Proceedings of the London Mathematical Society. Third Series},
volume = {109},
year = {2014},
number = {6},
pages = {1402--1434},
issn = {0024-6115},
mrclass = {11G35 (14G05)},
mrnumber = {3293154},
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zblnumber = {06385144},
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@book{wittlnm, mrkey = {2307807},
author = {Wittenberg, Olivier},
title = {Intersections de deux quadriques et pinceaux de courbes de genre 1/{I}ntersections of two quadrics and pencils of curves of genus~1},
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volume = {1901},
publisher = {Springer-Verlag},
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pages = {viii+218},
isbn = {978-3-540-69137-2; 3-540-69137-5},
mrclass = {14G05 (11G05 14G25)},
mrnumber = {2307807},
mrreviewer = {Tam{á}s Szamuely},
doi = {10.1007/3-540-69137-5},
address = {New York},
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@article{wittdmj, mrkey = {2957699},
author = {Wittenberg, Olivier},
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year = {2012},
number = {11},
pages = {2113--2166},
issn = {0012-7094},
coden = {DUMJAO},
mrclass = {14C15 (14C25 14D10 14G25)},
mrnumber = {2957699},
mrreviewer = {Harry Tamvakis},
doi = {10.1215/00127094-1699441},
zblnumber = {1248.14030},
}