Algebraic families of nonzero elements of Shafarevich-Tate groups

Colliot-Thélène, Jean-Louis; Poonen, Bjorn

doi:10.1090/S0894-0347-99-00315-X

Algebraic families of nonzero elements of Shafarevich-Tate groups
HTML articles powered by AMS MathViewer

by Jean-Louis Colliot-Thélène and Bjorn Poonen

J. Amer. Math. Soc. 13 (2000), 83-99

DOI: https://doi.org/10.1090/S0894-0347-99-00315-X

Published electronically: August 20, 1999

PDF | Request permission

Abstract:

Principal homogeneous spaces under an abelian variety defined over a number field $k$ may have rational points in all completions of the number field without having rational points over $k$. Such principal homogeneous spaces represent the nonzero elements of the Shafarevich-Tate group of the abelian variety. We produce nontrivial, one-parameter families of such principal homogeneous spaces. The total space of these families are counterexamples to the Hasse principle. We show these may be accounted for by the Brauer-Manin obstruction.

References

M. Amer, Quadratische Formen über Funktionenkörpern, Dissertation, Mainz, 1976.
Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Armand Brumer, Remarques sur les couples de formes quadratiques, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 16, A679–A681 (French, with English summary). MR 498374
J. W. S. Cassels and M. J. T. Guy, On the Hasse principle for cubic surfaces, Mathematika 13 (1966), 111–120. MR 211966, DOI 10.1112/S0025579300003879
J.-L. Colliot-Thélène, The Hasse principle in a pencil of algebraic varieties, Number theory (Tiruchirapalli, 1996) Contemp. Math., vol. 210, Amer. Math. Soc., Providence, RI, 1998, pp. 19–39. MR 1478483, DOI 10.1090/conm/210/02782
J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 223–237 (French). MR 605344
Jean-Louis Colliot-Thélène, Dimitri Kanevsky, and Jean-Jacques Sansuc, Arithmétique des surfaces cubiques diagonales, Diophantine approximation and transcendence theory (Bonn, 1985) Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 1–108 (French). MR 927558, DOI 10.1007/BFb0078705
Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. I, J. Reine Angew. Math. 373 (1987), 37–107. MR 870307
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240, DOI 10.1007/BF02684599
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
A. Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique. V. Les schémas de Picard: théorèmes d’existence, in Séminaire Bourbaki, Vol. 7, Exp. 232, 143–161, Soc. Math. France, Paris, 1995.
Alexander Grothendieck, Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 46–66 (French). MR 244269
David Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75 (1994), no. 1, 221–260 (French). MR 1284820, DOI 10.1215/S0012-7094-94-07507-8
D. Harari, Weak approximation and non-abelian fundamental groups, preprint, October 1998.
D. Harari and A. N. Skorobogatov, Non-abelian cohomology and rational points, preprint, May 1999.
Uwe Jannsen, Principe de Hasse cohomologique, Séminaire de Théorie des Nombres, Paris, 1989–90, Progr. Math., vol. 102, Birkhäuser Boston, Boston, MA, 1992, pp. 121–140 (French). MR 1476733, DOI 10.1007/978-1-4757-4269-5_{1}0
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
Stephen Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120–136. MR 242831, DOI 10.1007/BF01389795
C.-E. Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, Thesis, University of Uppsala, 1940.
Qing Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577–4610 (French, with English summary). MR 1363944, DOI 10.1090/S0002-9947-96-01684-4
Y. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 401–411. MR 0427322
Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, to appear in Ann. of Math.
Albert Eagle, Series for all the roots of a trinomial equation, Amer. Math. Monthly 46 (1939), 422–425. MR 5, DOI 10.2307/2303036
Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819, DOI 10.1007/BFb0074269
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
SGA 4, Théorie des topos et cohomologie étale des schémas, Tome 1. Théorie des topos. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964. Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972.
A. N. Skorobogatov, Beyond the Manin obstruction, Invent. Math. 135 (1999), no. 2, 399–424.
H. P. F. Swinnerton-Dyer, Two special cubic surfaces, Mathematika 9 (1962), 54–56. MR 139989, DOI 10.1112/S0025579300003090
Peter Swinnerton-Dyer, The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc. 113 (1993), no. 3, 449–460. MR 1207510, DOI 10.1017/S0305004100076106
J. Van Geel and V. I. Yanchevskii, Indices of hyperelliptic curves over $p$-adic fields, Manuscripta Math. 96 (1998), no. 3, 317–333. MR 1638165, DOI 10.1007/s002290050070