Abstract
The Manin conjecture is established for Châtelet surfaces over $\mathbf{Q}$ arising as minimal proper smooth models of the surface $Y^2+Z^2=f(X)$ in $\mathbf{A}_{\mathbf{Q}}^3$, where $f\in \mathbf{Z}[X]$ is a totally reducible polynomial of degree $3$ without repeated roots. These surfaces do not satisfy weak approximation.
-
[batyrevmanin:hauteur]
V. V. Batyrev and Y. I. Manin, "Sur le nombre des points rationnels de hauteur borné des variétés algébriques," Math. Ann., vol. 286, iss. 1-3, pp. 27-43, 1990.
@article {batyrevmanin:hauteur, MRKEY = {1032922},
AUTHOR = {Batyrev, V. V. and Manin, Yu. I.},
TITLE = {Sur le nombre des points rationnels de hauteur borné des variétés algébriques},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {286},
YEAR = {1990},
NUMBER = {1-3},
PAGES = {27--43},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11G35 (14G10 14G40)},
MRNUMBER = {1032922},
MRREVIEWER = {Marc Hindry},
DOI = {10.1007/BF01453564},
ZBLNUMBER = {0679.14008},
} -
[batyrevtschinkel:toric]
V. V. Batyrev and Y. Tschinkel, "Rational points of bounded height on compactifications of anisotropic tori," Internat. Math. Res. Notices, iss. 12, pp. 591-635, 1995.
@article {batyrevtschinkel:toric, MRKEY = {1369408},
AUTHOR = {Batyrev, Victor V. and Tschinkel, Yuri},
TITLE = {Rational points of bounded height on compactifications of anisotropic tori},
JOURNAL = {Internat. Math. Res. Notices},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {1995},
NUMBER = {12},
PAGES = {591--635},
ISSN = {1073-7928},
MRCLASS = {14G05 (11G35 14M25)},
MRNUMBER = {1369408},
MRREVIEWER = {Emmanuel Peyre},
DOI = {10.1155/S1073792895000365},
ZBLNUMBER = {0890.14008},
} -
[bretechebrowning:sums]
R. de la Bretèche and T. D. Browning, "Sums of arithmetic functions over values of binary forms," Acta Arith., vol. 125, iss. 3, pp. 291-304, 2006.
@article {bretechebrowning:sums, MRKEY = {2276196},
AUTHOR = {de la Bret{è}che, R. and Browning, T. D.},
TITLE = {Sums of arithmetic functions over values of binary forms},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {125},
YEAR = {2006},
NUMBER = {3},
PAGES = {291--304},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11N37 (11N32)},
MRNUMBER = {2276196},
MRREVIEWER = {Ognian Trifonov},
DOI = {10.4064/aa125-3-6},
ZBLNUMBER = {1159.11035},
} -
[bretechebrowning:4linear]
R. de la Bretèche and T. D. Browning, "Binary linear forms as sums of two squares," Compos. Math., vol. 144, iss. 6, pp. 1375-1402, 2008.
@article {bretechebrowning:4linear, MRKEY = {2474314},
AUTHOR = {de la Bret{è}che, R. and Browning, T. D.},
TITLE = {Binary linear forms as sums of two squares},
JOURNAL = {Compos. Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {144},
YEAR = {2008},
NUMBER = {6},
PAGES = {1375--1402},
ISSN = {0010-437X},
MRCLASS = {11N37 (11E16 11N25)},
MRNUMBER = {2474314},
MRREVIEWER = {R. A. Mollin},
DOI = {10.1112/S0010437X08003692},
ZBLNUMBER = {05382617},
} -
[chatelet:points] F. Châtelet, "Points rationnels sur certaines courbes et surfaces cubiques," Enseignement Math., vol. 5, pp. 153-170 (1960), 1959.
@article {chatelet:points, MRKEY = {0130218},
AUTHOR = {Ch{â}telet, Fran{ç}ois},
TITLE = {Points rationnels sur certaines courbes et surfaces cubiques},
JOURNAL = {Enseignement Math.},
FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
VOLUME = {5},
YEAR = {1959},
PAGES = {153--170 (1960)},
ISSN = {0013-8584},
MRCLASS = {14.40 (14.49)},
MRNUMBER = {0130218},
MRREVIEWER = {J. W. S. Cassels},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0100.27404},
} -
[chatelet:points2] F. Châtelet, "Points rationnels sur certaines surfaces cubiques," in Les Tendances Géom. en Algèbre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 67-75.
@incollection {chatelet:points2, MRKEY = {0213353},
AUTHOR = {Ch{â}telet, Fran{ç}ois},
TITLE = {Points rationnels sur certaines surfaces cubiques},
BOOKTITLE = {Les {T}endances {G}éom. en {A}lgèbre et {T}héorie des {N}ombres},
PAGES = {67--75},
PUBLISHER = {Éditions du Centre National de la Recherche Scientifique, Paris},
YEAR = {1966},
MRCLASS = {14.40},
MRNUMBER = {0213353},
MRREVIEWER = {T. Ono},
ZBLNUMBER = {0178.24101},
} -
[cts:predescente2] J. Colliot-Thélène and J. Sansuc, "La descente sur une variété rationnelle définie sur un corps de nombres," C. R. Acad. Sci. Paris Sér. A-B, vol. 284, iss. 19, p. a1215-a1218, 1977.
@article {cts:predescente2, MRKEY = {0447250},
AUTHOR = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques},
TITLE = {La descente sur une variété rationnelle définie sur un corps de nombres},
JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
VOLUME = {284},
YEAR = {1977},
NUMBER = {19},
PAGES = {A1215--A1218},
MRCLASS = {14G25 (14M20)},
MRNUMBER = {0447250},
ZBLNUMBER={0349.14015},
} -
[cts:descente1] J. Colliot-Thélène and J. Sansuc, "La descente sur les variétés rationnelles," in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980, pp. 223-237.
@incollection {cts:descente1, MRKEY = {0605344},
AUTHOR = {Colliot-Th{é}l{è}ne, Jean-Louis and Sansuc, Jean-Jacques},
TITLE = {La descente sur les variétés rationnelles},
BOOKTITLE = {Journées de {G}éometrie {A}lgébrique d'{A}ngers, {J}uillet 1979/{A}lgebraic {G}eometry, {A}ngers, 1979},
PAGES = {223--237},
PUBLISHER = {Sijthoff \& Noordhoff},
ADDRESS = {Alphen aan den Rijn},
YEAR = {1980},
MRCLASS = {14G05 (14C15)},
MRNUMBER = {0605344},
MRREVIEWER = {Daniel Coray},
ZBLNUMBER = {0451.14018},
} -
[cts:descente2]
J. Colliot-Thélène and J. Sansuc, "La descente sur les variétés rationnelles. II," Duke Math. J., vol. 54, iss. 2, pp. 375-492, 1987.
@article {cts:descente2, MRKEY = {0899402},
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.},
FJOURNAL = {Duke Mathematical Journal},
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},
} -
[ctssd:chatelet1] 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 {ctssd:chatelet1, 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 = {0622.14029},
} -
[ctssd:chatelet2] 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 {ctssd:chatelet2, MRKEY = {0876222},
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 = {0876222},
MRREVIEWER = {Noriko Yui},
ZBLNUMBER = {0622.14030},
} -
[cook:quadratic] R. J. Cook, "Simultaneous quadratic equations," J. London Math. Soc., vol. 4, pp. 319-326, 1971.
@article {cook:quadratic, MRKEY = {0289406},
AUTHOR = {Cook, R. J.},
TITLE = {Simultaneous quadratic equations},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {4},
YEAR = {1971},
PAGES = {319--326},
ISSN = {0024-6107},
MRCLASS = {10.10},
MRNUMBER = {0289406},
MRREVIEWER = {T. T. Tonkov},
ZBLNUMBER = {0224.10021},
} -
[coraytsfasman:delpezzo]
D. F. Coray and M. A. Tsfasman, "Arithmetic on singular Del Pezzo surfaces," Proc. London Math. Soc., vol. 57, iss. 1, pp. 25-87, 1988.
@article {coraytsfasman:delpezzo, MRKEY = {0940430},
AUTHOR = {Coray, D. F. and Tsfasman, M. A.},
TITLE = {Arithmetic on singular {D}el {P}ezzo surfaces},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {57},
YEAR = {1988},
NUMBER = {1},
PAGES = {25--87},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {11G35 (11E04 11E95 11G25 14J17 14J20 14J26)},
MRNUMBER = {0940430},
MRREVIEWER = {Noriko Yui},
DOI = {10.1112/plms/s3-57.1.25},
ZBLNUMBER = {0653.14018},
} -
[davenport:cubic] H. Davenport, "Cubic forms in sixteen variables," Proc. Roy. Soc. Ser. A, vol. 272, pp. 285-303, 1963.
@article {davenport:cubic, MRKEY = {0155800},
AUTHOR = {Davenport, H.},
TITLE = {Cubic forms in sixteen variables},
JOURNAL = {Proc. Roy. Soc. Ser. A},
VOLUME = {272},
YEAR = {1963},
PAGES = {285--303},
MRCLASS = {10.17},
MRNUMBER = {0155800},
MRREVIEWER = {B. J. Birch},
ZBLNUMBER = {0107.04102},
} -
[draxl:tori] P. K. J. Draxl, "$L$-Funktionen algebraischer Tori," J. Number Theory, vol. 3, pp. 444-467, 1971.
@article {draxl:tori, MRKEY = {0291099},
AUTHOR = {Draxl, P. K. J.},
TITLE = {{$L$}-{F}unktionen algebraischer {T}ori},
JOURNAL = {J. Number Theory},
FJOURNAL = {Journal of Number Theory},
VOLUME = {3},
YEAR = {1971},
PAGES = {444--467},
ISSN = {0022-314X},
MRCLASS = {10H10},
MRNUMBER = {0291099},
MRREVIEWER = {T. Ono},
ZBLNUMBER = {0231.12018},
} -
[heathbrown:2003] D. R. Heath-Brown, "Linear relations amongst sums of two squares," in Number Theory and Algebraic Geometry, Cambridge: Cambridge Univ. Press, 2003, vol. 303, pp. 133-176.
@incollection {heathbrown:2003, MRKEY = {2053459},
AUTHOR = {Heath-Brown, D. R.},
TITLE = {Linear relations amongst sums of two squares},
BOOKTITLE = {Number Theory and Algebraic Geometry},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {303},
PAGES = {133--176},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2003},
MRCLASS = {11N25},
MRNUMBER = {2053459},
MRREVIEWER = {G. Greaves},
ZBLNUMBER = {1161.11387},
} -
[heathbrownskorobogatov]
R. Heath-Brown and A. Skorobogatov, "Rational solutions of certain equations involving norms," Acta Math., vol. 189, iss. 2, pp. 161-177, 2002.
@article {heathbrownskorobogatov, MRKEY = {1961196},
AUTHOR = {Heath-Brown, Roger and Skorobogatov, Alexei},
TITLE = {Rational solutions of certain equations involving norms},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {189},
YEAR = {2002},
NUMBER = {2},
PAGES = {161--177},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {14G05 (11D57 11G25 14F22 14G25)},
MRNUMBER = {1961196},
MRREVIEWER = {Tam{á}s Szamuely},
DOI = {10.1007/BF02392841},
ZBLNUMBER = {1023.11033},
} -
[iskovskih:hasse] 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 {iskovskih:hasse, 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 = {0286743},
MRREVIEWER = {G. Maxwell},
ZBLNUMBER={0232.10015},
} -
[lachaud:waring] G. Lachaud, "Une présentation adélique de la série singulière et du problème de Waring," Enseign. Math., vol. 28, iss. 1-2, pp. 139-169, 1982.
@article {lachaud:waring, MRKEY = {0670501},
AUTHOR = {Lachaud, Gilles},
TITLE = {Une présentation adélique de la série singulière et du problème de {W}aring},
JOURNAL = {Enseign. Math.},
FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
VOLUME = {28},
YEAR = {1982},
NUMBER = {1-2},
PAGES = {139--169},
ISSN = {0013-8584},
CODEN = {ENMAAR},
MRCLASS = {10J10},
MRNUMBER = {0670501},
MRREVIEWER = {T. Ono},
ZBLNUMBER = {0499.12010},
} -
@article {ono:tori, MRKEY = {0124326},
AUTHOR = {Ono, Takashi},
TITLE = {Arithmetic of algebraic tori},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {74},
YEAR = {1961},
PAGES = {101--139},
ISSN = {0003-486X},
MRCLASS = {14.48 (14.49)},
MRNUMBER = {0124326},
MRREVIEWER = {T. Nakayama},
ZBLNUMBER = {0119.27801},
DOI={10.2307/1970307},
} -
[ono:tamagawa]
T. Ono, "On the Tamagawa number of algebraic tori," Ann. of Math., vol. 78, pp. 47-73, 1963.
@article {ono:tamagawa, MRKEY = {0156851},
AUTHOR = {Ono, Takashi},
TITLE = {On the {T}amagawa number of algebraic tori},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {78},
YEAR = {1963},
PAGES = {47--73},
ISSN = {0003-486X},
MRCLASS = {14.50 (10.16)},
MRNUMBER = {0156851},
MRREVIEWER = {A. Mattuck},
ZBLNUMBER = {0122.39101},
DOI={10.2307/1970502},
} -
[peyre:fano]
E. Peyre, "Hauteurs et mesures de Tamagawa sur les variétés de Fano," Duke Math. J., vol. 79, iss. 1, pp. 101-218, 1995.
@article {peyre:fano, MRKEY = {1340296},
AUTHOR = {Peyre, Emmanuel},
TITLE = {Hauteurs et mesures de {T}amagawa sur les variétés de {F}ano},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {79},
YEAR = {1995},
NUMBER = {1},
PAGES = {101--218},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11G35 (14G05 14J45)},
MRNUMBER = {1340296},
MRREVIEWER = {Shouwu Zhang},
DOI = {10.1215/S0012-7094-95-07904-6},
ZBLNUMBER = {0901.14025},
} -
[peyre:torseurs] E. Peyre, "Terme principal de la fonction zêta des hauteurs et torseurs universels," in Nombre et Répartition de Points de Hauteur Bornée, , 1998, vol. 251, pp. 259-298.
@incollection {peyre:torseurs, MRKEY = {1679842},
AUTHOR = {Peyre, Emmanuel},
TITLE = {Terme principal de la fonction zêta des hauteurs et torseurs universels},
BOOKTITLE={Nombre et R{é}partition de Points de Hauteur Born{é}e},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {251},
YEAR = {1998},
PAGES = {259--298},
ISSN = {0303-1179},
MRCLASS = {11G50 (14G05 14G10)},
MRNUMBER = {1679842},
MRREVIEWER = {Carlo Gasbarri},
ZBLNUMBER = {0966.14016},
} -
[peyre:cercle] E. Peyre, "Torseurs universels et méthode du cercle," in Rational Points on Algebraic Varieties, Basel: Birkhäuser, 2001, vol. 199, pp. 221-274.
@incollection {peyre:cercle, MRKEY = {1875176},
AUTHOR = {Peyre, Emmanuel},
TITLE = {Torseurs universels et méthode du cercle},
BOOKTITLE = {Rational Points on Algebraic Varieties},
SERIES = {Progr. Math.},
VOLUME = {199},
PAGES = {221--274},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
MRCLASS = {11G35 (11G50 14G05 14G40)},
MRNUMBER = {1875176},
MRREVIEWER = {Matthias Strauch},
ZBLNUMBER = {1092.14506},
} -
[peyre:lille]
E. Peyre, "Points de hauteur bornée, topologie adélique et mesures de Tamagawa," J. Théor. Nombres Bordeaux, vol. 15, iss. 1, pp. 319-349, 2003.
@article {peyre:lille, MRKEY = {2019019},
AUTHOR = {Peyre, Emmanuel},
TITLE = {Points de hauteur bornée, topologie adélique et mesures de {T}amagawa},
JOURNAL = {J. Théor. Nombres Bordeaux},
FJOURNAL = {Journal de Théorie des Nombres de Bordeaux},
VOLUME = {15},
YEAR = {2003},
NUMBER = {1},
PAGES = {319--349},
ISSN = {1246-7405},
MRCLASS = {14G05 (11G35 11G50 14G25)},
MRNUMBER = {2019019},
MRREVIEWER = {Timothy D. Browning},
DOI={10.5802/jtnb.405},
ZBLNUMBER = {1057.14031},
} -
[salberger:tamagawa] P. Salberger, "Tamagawa measures on universal torsors and points of bounded height on Fano varieties," in Nombre et Répartition de Points de Hauteur Bornée, , 1998, vol. 251, pp. 91-258.
@incollection {salberger:tamagawa, MRKEY = {1679841},
AUTHOR = {Salberger, Per},
TITLE = {Tamagawa measures on universal torsors and points of bounded height on {F}ano varieties},
BOOKTITLE= {Nombre et R{é}partition de Points de Hauteur Born{é}e},
JOURNAL = {Astérisque},
FJOURNAL = {Astérisque},
VOLUME = {251},
YEAR = {1998},
PAGES = {91--258},
ISSN = {0303-1179},
MRCLASS = {11G50 (11G35 14G05 14G25 14J45)},
MRNUMBER = {1679841},
MRREVIEWER = {David Harari},
ZBLNUMBER = {0959.14007},
} -
[skorobogatov:torsors]
A. Skorobogatov, Torsors and Rational Points, Cambridge: Cambridge Univ. Press, 2001, vol. 144.
@book {skorobogatov:torsors, MRKEY = {1845760},
AUTHOR = {Skorobogatov, Alexei},
TITLE = {Torsors and Rational Points},
SERIES = {Cambridge Tracts in Math.},
VOLUME = {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 = {1845760},
MRREVIEWER = {Tam{á}s Szamuely},
DOI = {10.1017/CBO9780511549588},
ZBLNUMBER = {0972.14015},
} -
[weil:adeles] A. Weil, Adeles and Algebraic Groups, Mass.: Birkhäuser, 1982, vol. 23.
@book {weil:adeles, MRKEY = {0670072},
AUTHOR = {Weil, Andr{é}},
TITLE = {Adeles and Algebraic Groups},
SERIES = {Progr. Math.},
VOLUME = {23},
PUBLISHER = {Birkhäuser},
ADDRESS = {Mass.},
YEAR = {1982},
PAGES = {iii+126},
ISBN = {3-7643-3092-9},
MRCLASS = {10C30 (12A82 12A85 20G35)},
MRNUMBER = {0670072},
MRREVIEWER = {K. F. Lai},
ZBLNUMBER = {0493.14028},
}