On the order three Brauer classes for cubic surfaces

• Autorzy: Andreas-Stephan Elsenhans , Jörg Jahnel
• Języki publikacji: EN

Abstrakty

EN

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.

Słowa kluczowe

EN

Cubic surface; Steiner trihedron; Triplet; Twisted cubic curve; Weak approximation; Explicit Brauer-Manin obstruction

Kategorie tematyczne

• 11D25: Cubic and quartic equations
• 14J26: Rational and ruled surfaces
• 11D85: Representation problems
• 14J20: Arithmetic ground fields

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 3
• Strony: 903-926

Daty

wydano

2012-06-01

online

2012-03-24

Twórcy

autor

Andreas-Stephan Elsenhans

• Universität Bayreuth

autor

Jörg Jahnel

• Universität Siegen

Bibliografia

• [1] Buckley A., Košir T., Determinantal representations of smooth cubic surfaces, Geom. Dedicata, 2007, 125, 115–140 http://dx.doi.org/10.1007/s10711-007-9144-x
• [2] Cartan H., Eilenberg S., Homological Algebra, Princeton University Press, Princeton, 1956
• [3] Cassels J.W.S., Guy M.J.T., On the Hasse principle for cubic surfaces, Mathematika, 1966, 13, 111–120 http://dx.doi.org/10.1112/S0025579300003879
• [4] Clebsch A., Die Geometrie auf den Flächen dritter Ordnung, J. Reine Angew. Math., 1866, 1866(65), 359–380 http://dx.doi.org/10.1515/crll.1866.65.359
• [5] Colliot-Thélène J.-L., Kanevsky D., Sansuc J.-J., Arithmétique des surfaces cubiques diagonales, In: Diophantine Approximation and Transcendence Theory, Bonn, May–June, 1985, Lecture Notes in Math., 1290, Springer, Berlin, 1987, 1–108 http://dx.doi.org/10.1007/BFb0078705
• [6] Cremona L., Sulle ventisette rette della superficie del terzo ordine, Istit. Lombardo Accad. Sci. Lett. Rend. A, 1870, 3, 209–219
• [7] Dolgachev I.V., Classical Algebraic Geometry, Cambridge University Press (in press), available at http://www.math.lsa.umich.edu/_idolga/CAG.pdf
• [8] Elsenhans A.-S., Jahnel J., Experiments with general cubic surfaces, In: Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, 1, Progr. Math., 269, Birkhäuser, Boston, 2007, 637–653
• [9] Elsenhans A.-S., Jahnel J., On the Brauer-Manin obstruction for cubic surfaces, J. Comb. Number Theory, 2010, 2(2), 107–128
• [10] Elsenhans A.-S., Jahnel J., Cubic surfaces with a Galois invariant pair of Steiner trihedra, Int. J. Number Theory, 2011, 7(4), 947–970 http://dx.doi.org/10.1142/S1793042111004253
• [11] Elsenhans A.-S., Jahnel J., On the quasi group of a cubic surface over a finite field, J. Number Theory (in press)
• [12] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, New York-Heidelberg, 1977
• [13] Henderson A., The Twenty-Seven Lines upon the Cubic Surface, Hafner, New York, 1960
• [14] Jahnel J., More cubic surfaces violating the Hasse principle, J. Théor. Nombres Bordeaux, 2011, 23(2), 471–477 http://dx.doi.org/10.5802/jtnb.772
• [15] Kresch A., Tschinkel Yu., On the arithmetic of del Pezzo surfaces of degree 2, Proc. London Math. Soc., 2004, 89(3), 545–569 http://dx.doi.org/10.1112/S002461150401490X
• [16] Kunyavskiĭ B.È., Skorobogatov A.N., Tsfasman M.A., Del Pezzo Surfaces of Degree Four, Mém. Soc. Math. France (N.S.), 37, Société Mathématique de France, Marseille, 1989
• [17] Manin Yu.I., Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Math. Library, 4, North-Holland/Elsevier, Amsterdam-London/New York, 1974
• [18] Mordell L.J., On the conjecture for the rational points on a cubic surface, J. London Math. Soc., 1965, 40, 149–158 http://dx.doi.org/10.1112/jlms/s1-40.1.149
• [19] Neukirch J., Klassenkörpertheorie, 2nd ed., Springer-Lehrbuch, Springer, Berlin, 2011 http://dx.doi.org/10.1007/978-3-642-17325-7
• [20] Peyre E., Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J., 1995, 79(1), 101–218 http://dx.doi.org/10.1215/S0012-7094-95-07904-6
• [21] Peyre E., Tschinkel Yu., Tamagawa numbers of diagonal cubic surfaces, numerical evidence, Math. Comp., 2001, 70(233), 367–387 http://dx.doi.org/10.1090/S0025-5718-00-01189-3
• [22] Schläfli L., An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface, The Quarterly Journal of Pure and Applied Mathematics, 1858, 2, 110–120
• [23] Serre J.-P., Local class field theory, In: Algebraic Number Theory, Brighton, 1965, Thompson, Washington, 1967, 128–161
• [24] Steiner J., Über die Flächen dritten Grades, J. Reine Angew. Math., 1857, 53, 133–141 http://dx.doi.org/10.1515/crll.1857.53.133
• [25] Swinnerton-Dyer H.P.F., Two special cubic surfaces, Mathematika, 1962, 9, 54–56 http://dx.doi.org/10.1112/S0025579300003090
• [26] Swinnerton-Dyer H.P.F., Universal equivalence for cubic surfaces over finite and local fields, In: Symposia Mathematica, 24, Rome, April 9–14, 1979, Academic Press, London-New York, 1981, 111–143
• [27] Swinnerton-Dyer P., The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc., 1993, 113(3), 449–460 http://dx.doi.org/10.1017/S0305004100076106
• [28] Tate J.T., Global class field theory, In: Algebraic Number Theory, Brighton, 1965, Thompson, Washington, 1967, 162–203

Identyfikatory

DOI

10.2478/s11533-012-0042-6