Abstract
Let $k$ be any global field of characteristic not $2$. We construct a $k$-variety $X$ such that $X(k)$ is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite étale covers.
-
[Colliot-Thelene1997] J. Colliot-Thélène, "Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale," in Algebraic $K$-Theory, Providence, RI, 1999, pp. 1-12.
@inproceedings{Colliot-Thelene1997,
author = {Colliot-Th{é}l{è}ne, Jean-Louis},
TITLE={Conjectures de type local-global sur l'image des groupes de Chow dans la cohomologie étale},
language={French},
BOOKTITLE={Algebraic $K$-Theory},
VENUE={Seattle, WA, 1997},
SERIES={Proc. Sympos. Pure Math.},
VOLUME={67},
YEAR={1999},
PAGES={1--12},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
MRNUMBER={1743234},
} -
[Colliot-Thelene2009preprint] J. Colliot-Thélène, "Zéro-cycles de degré 1 sur les solides de Poonen," , preprint , 2009.
@techreport{Colliot-Thelene2009preprint,
author = {Colliot-Th{é}l{è}ne, Jean-Louis},
TITLE={Zéro-cycles de degré 1 sur les solides de Poonen},
YEAR={2009},
TYPE={preprint},
NOTE={{\it Bull. Math. Soc. France},
to appear},
} -
[CT-Sansuc-SD1987I] 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 {CT-Sansuc-SD1987I, MRKEY = {870307},
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 = {88m:11045a},
MRREVIEWER = {Noriko Yui},
ZBLNUMBER = {0622.14029},
} -
[CT-Sansuc-SD1987II] J. Colliot-Thélène, J. Sansuc, and P. Swinnerton-Dyer, "Intersections of two quadrics and Châtelet surfaces. II," J. Reine Angew. Math., vol. 374, pp. 72-168, 1987.
@article {CT-Sansuc-SD1987II, MRKEY = {876222},
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. {II}},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {374},
YEAR = {1987},
PAGES = {72--168},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11G35 (11E81 14J20 14J26)},
MRNUMBER = {88m:11045b},
MRREVIEWER = {Noriko Yui},
ZBLNUMBER = {0622.14030},
} -
[Demarche2009]
C. Demarche, "Obstruction de descente et obstruction de Brauer-Manin étale," Algebra Number Theory, vol. 3, iss. 2, pp. 237-254, 2009.@article {Demarche2009, MRKEY = {2491944},
AUTHOR = {Demarche, Cyril},
TITLE = {Obstruction de descente et obstruction de {B}rauer-{M}anin étale},
JOURNAL = {Algebra Number Theory},
FJOURNAL = {Algebra \& Number Theory},
VOLUME = {3},
YEAR = {2009},
NUMBER = {2},
PAGES = {237--254},
ISSN = {1937-0652},
MRCLASS = {11G35 (11E72 14Fxx 14G05)},
MRNUMBER = {2491944},
DOI = {10.2140/ant.2009.3.237},
ZBLNUMBER = {05566608},
} -
[Grothendieck-Brauer3] A. Grothendieck, "Le groupe de Brauer. III. Exemples et compléments," in Dix Exposés sur la Cohomologie des Schémas, Amsterdam: North-Holland, 1968, pp. 88-188.
@incollection {Grothendieck-Brauer3, MRKEY = {0244271},
AUTHOR = {Grothendieck, Alexander},
TITLE = {Le groupe de {B}rauer. {III}. {E}xemples et compléments},
BOOKTITLE = {Dix {E}xposés sur la {C}ohomologie des {S}chémas},
PAGES = {88--188},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1968},
MRCLASS = {14.55},
MRNUMBER = {39 \#5586c},
MRREVIEWER = {J. S. Milne},
ZBLNUMBER={0198.25901},
} -
[Harari2002] D. Harari, "Groupes algébriques et points rationnels," Math. Ann., vol. 322, iss. 4, pp. 811-826, 2002.
@article {Harari2002, MRKEY = {1905103},
AUTHOR = {Harari, David},
TITLE = {Groupes algébriques et points rationnels},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {322},
YEAR = {2002},
NUMBER = {4},
PAGES = {811--826},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14L99 (11E72 14F22)},
MRNUMBER = {2003e:14038},
MRREVIEWER = {Antoine Ducros},
DOI = {10.1007/s002080100289},
ZBLNUMBER = {1042.14004},
} -
[Hartshorne1977] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
@book {Hartshorne1977, MRKEY = {0463157},
AUTHOR = {Hartshorne, Robin},
TITLE = {Algebraic Geometry},
SERIES = {Graduate Texts in Mathematics},
VOLUME={52},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1977},
PAGES = {xvi+496},
ISBN = {0-387-90244-9},
MRCLASS = {14-01},
MRNUMBER = {57 \#3116},
MRREVIEWER = {Robert Speiser},
ZBLNUMBER = {0367.14001},
} -
[Iskovskikh1971] V. A. Iskovskih, "A counterexample to the Hasse principle for systems of two quadratic forms in five variables," Mat. Zametki, vol. 10, pp. 253-257, 1971.
@article {Iskovskikh1971, MRKEY = {0286743},
AUTHOR = {Iskovskih, V. A.},
TITLE = {A counterexample to the {H}asse principle for systems of two quadratic forms in five variables},
JOURNAL = {Mat. Zametki},
FJOURNAL = {Akademiya Nauk SSSR. Matematicheskie Zametki},
VOLUME = {10},
YEAR = {1971},
PAGES = {253--257},
ISSN = {0025-567X},
MRCLASS = {10.10 (14.00)},
MRNUMBER = {44 \#3952},
MRREVIEWER = {G. Maxwell},
ZBLNUMBER={0232.10015},
} -
[Lind1940] C. Lind, "Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins," Thesis, University of Uppsala,, vol. 1940, p. 97, 1940.
@article {Lind1940, MRKEY = {0022563},
AUTHOR = {Lind, Carl-Erik},
TITLE = {Untersuchungen über die rationalen {P}unkte der ebenen kubischen {K}urven vom {G}eschlecht {E}ins},
JOURNAL = {Thesis, University of Uppsala,},
VOLUME = {1940},
YEAR = {1940},
PAGES = {97},
MRCLASS = {10.0X},
MRNUMBER = {9,225c},
MRREVIEWER = {A. Weil},
JFMNUMBER={66.0131.04},
} -
[Manin1971] Y. I. Manin, "Le groupe de Brauer-Grothendieck en géométrie diophantienne," in Actes du Congrès International des Mathématiciens , Tome 1, Paris: Gauthier-Villars, 1971, pp. 401-411.
@incollection {Manin1971, MRKEY = {0427322},
AUTHOR = {Manin, Y. I.},
TITLE = {Le groupe de {B}rauer-{G}rothendieck en géométrie diophantienne},
BOOKTITLE = {Actes du {C}ongrès {I}nternational des {M}athématiciens , {T}ome {\rm 1}},
VENUE={{N}ice, 1970},
PAGES = {401--411},
PUBLISHER = {Gauthier-Villars},
ADDRESS = {Paris},
YEAR = {1971},
MRCLASS = {14G25},
MRNUMBER = {55 \#356},
MRREVIEWER = {Loren D. Olson},
ZBLNUMBER = {0239.14010},
} -
[Poonen-complete-intersection2001] B. Poonen, "The Hasse principle for complete intersections in projective space," in Rational Points on Algebraic Varieties, Basel: Birkhäuser, 2001, pp. 307-311.
@incollection {Poonen-complete-intersection2001, MRKEY = {1875178},
AUTHOR = {Poonen, Bjorn},
TITLE = {The {H}asse principle for complete intersections in projective space},
BOOKTITLE = {Rational Points on Algebraic Varieties},
SERIES = {Progr. Math.},
NUMBER = {199},
PAGES = {307--311},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
MRCLASS = {14G25 (11G35)},
MRNUMBER = {2002j:14028},
MRREVIEWER = {Antoine Ducros},
ZBLNUMBER = {1079.14027},
} -
[Poonen2009-chatelet] B. Poonen, "Existence of rational points on smooth projective varieties," J. Eur. Math. Soc. $($JEMS$)$, vol. 11, iss. 3, pp. 529-543, 2009.
@article {Poonen2009-chatelet, MRKEY = {2505440},
AUTHOR = {Poonen, Bjorn},
TITLE = {Existence of rational points on smooth projective varieties},
JOURNAL = {J. Eur. Math. Soc. $($JEMS$)$},
FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
VOLUME = {11},
YEAR = {2009},
NUMBER = {3},
PAGES = {529--543},
ISSN = {1435-9855},
MRCLASS = {14G05 (11G35 11U05)},
MRNUMBER = {2010c:14019},
MRREVIEWER = {Timothy D. Browning},
DOI = {10.4171/JEMS/159},
ZBLNUMBER = {05565390},
} -
[Reichardt1942] H. Reichardt, "Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen," J. Reine Angew. Math., vol. 184, pp. 12-18, 1942.
@article {Reichardt1942, MRKEY = {0009381},
AUTHOR = {Reichardt, Hans},
TITLE = {Einige im {K}leinen überall lösbare, im {G}rossen unlösbare diophantische {G}leichungen},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {184},
YEAR = {1942},
PAGES = {12--18},
ISSN = {0075-4102},
MRCLASS = {10.0X},
MRNUMBER = {5,141c},
MRREVIEWER = {A. Brauer},
DOI = {10.1515/crll.1942.184.12},
ZBLNUMBER={0026.29701},
} -
[Sarnak-Wang1995] P. Sarnak and L. Wang, "Some hypersurfaces in ${\bf P}^4$ and the Hasse-principle," C. R. Acad. Sci. Paris Sér. I Math., vol. 321, iss. 3, pp. 319-322, 1995.
@article {Sarnak-Wang1995, MRKEY = {1346134},
AUTHOR = {Sarnak, Peter and Wang, Lan},
TITLE = {Some hypersurfaces in {${\bf P}\sp 4$} and the {H}asse-principle},
JOURNAL = {C. R. Acad. Sci. Paris Sér. {\rm I} Math.},
FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {321},
YEAR = {1995},
NUMBER = {3},
PAGES = {319--322},
ISSN = {0764-4442},
CODEN = {CASMEI},
MRCLASS = {14G05 (11G35 14G20)},
MRNUMBER = {96j:14014},
MRREVIEWER = {Junjiro Noguchi},
ZBLNUMBER = {0857.14013},
} -
[SGA1] A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Paris: Société Mathématique de France, 2003.
@book {SGA1, KEY={SGA1},
author={Grothendieck, Alexander},
MRKEY = {2017446},
TITLE = {Revêtements étales et groupe fondamental {\rm ({SGA} 1)}},
SERIES = {Documents Mathématiques (Paris) {\rm [}Mathematical Documents (Paris){\rm ]},
3},
NOTE = {S{é}minaire de g{é}om{é}trie alg{é}brique du Bois Marie 1960--61. [Algebraic Geometry Seminar of Bois Marie 1960--61], Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [{\it Lecture Notes in Math.} {\bf 224},
Springer-Verlag, New York; MR0354651 (50 \#7129)]},
PUBLISHER = {Société Mathématique de France},
ADDRESS = {Paris},
YEAR = {2003},
PAGES = {xviii+327},
ISBN = {2-85629-141-4},
MRCLASS = {14E20 (14-06 14F35)},
MRNUMBER = {2004g:14017},
} -
[Skorobogatov1999] A. N. Skorobogatov, "Beyond the Manin obstruction," Invent. Math., vol. 135, iss. 2, pp. 399-424, 1999.
@article {Skorobogatov1999, MRKEY = {1666779},
AUTHOR = {Skorobogatov, Alexei N.},
TITLE = {Beyond the {M}anin obstruction},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {135},
YEAR = {1999},
NUMBER = {2},
PAGES = {399--424},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {14F22 (11G35 14F42 14G25)},
MRNUMBER = {2000c:14022},
MRREVIEWER = {Tam{á}s Szamuely},
DOI = {10.1007/s002220050291},
ZBLNUMBER = {0951.14013},
} -
[Skorobogatov2001] A. N. Skorobogatov, Torsors and Rational Points, Cambridge: Cambridge Univ. Press, 2001.
@book {Skorobogatov2001, MRKEY = {1845760},
AUTHOR = {Skorobogatov, Alexei N.},
TITLE = {Torsors and Rational Points},
SERIES = {Cambridge Tracts in Math.},
NUMBER = {144},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2001},
PAGES = {viii+187},
ISBN = {0-521-80237-7},
MRCLASS = {14G05 (11G35 11S25 14D10 14G25 14L30)},
MRNUMBER = {2002d:14032},
MRREVIEWER = {Tam{á}s Szamuely},
ZBLNUMBER = {0972.14015},
} -
[Skorobogatov2009] A. N. Skorobogatov, "Descent obstruction is equivalent to étale Brauer-Manin obstruction," Math. Ann., vol. 344, iss. 3, pp. 501-510, 2009.
@article {Skorobogatov2009, MRKEY = {2501300},
AUTHOR = {Skorobogatov, Alexei N.},
TITLE = {Descent obstruction is equivalent to étale {B}rauer-{M}anin obstruction},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {344},
YEAR = {2009},
NUMBER = {3},
PAGES = {501--510},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14G25 (14Fxx)},
MRNUMBER = {2501300},
DOI = {10.1007/s00208-008-0314-4},
ZBLNUMBER = {1180.14017},
} -
[Stoll2007] M. Stoll, "Finite descent obstructions and rational points on curves," Algebra Number Theory, vol. 1, iss. 4, pp. 349-391, 2007.
@article {Stoll2007, MRKEY = {2368954},
AUTHOR = {Stoll, Michael},
TITLE = {Finite descent obstructions and rational points on curves},
JOURNAL = {Algebra Number Theory},
FJOURNAL = {Algebra \& Number Theory},
VOLUME = {1},
YEAR = {2007},
NUMBER = {4},
PAGES = {349--391},
ISSN = {1937-0652},
MRCLASS = {11G30 (11G10 14G05)},
MRNUMBER = {2008i:11086},
MRREVIEWER = {Pavlos Tzermias},
DOI = {10.2140/ant.2007.1.349},
ZBLNUMBER = {1167.11024},
}
Full Article