Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T05:15:22.836Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONAL POINTS ON INTERSECTIONS OF CUBIC AND QUADRIC HYPERSURFACES

Part of: Arithmetic algebraic geometry Additive number theory; partitions Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  05 June 2014

T. D. Browning ,
R. Dietmann  and
D. R. Heath-Brown
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK ([email protected])
R. Dietmann
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, TW20 OEX, UK ([email protected])
D. R. Heath-Brown
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.

Keywords

Hasse principlecubicquadraticsystemrational pointDiophantine equationscircle methodWeyl sumvan der Corput methodcomplete intersection

MSC classification

Primary: 11G35: Varieties over global fields 11P55: Applications of the Hardy-Littlewood method 14G05: Rational points
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 4 , October 2015 , pp. 703 - 749
Copyright
© Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aznar, V. N., On the Chern classes and the Euler characteristic for non-singular complete intersections, Proc. Amer. Math. Soc. 78 (1980), 143–148.CrossRefGoogle Scholar
Baker, R. C., Diophantine inequalities, London Mathematical Society Monographs (Oxford University Press, 1986).Google Scholar
Birch, B. J., Forms in many variables, Proc. Roy. Soc. Ser. A 265 (1961/62), 245–263.Google Scholar
Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces, J. Reine Angew. Math. 629 (2009), 37–88.Google Scholar
Brüdern, J., Dietmann, R., Liu, J. Y. and Wooley, T.D., A Birch–Goldbach theorem, Arch. Math. (Basel) 94 (2010), 53–58.Google Scholar
Colliot-Thélène, J.-L., Coray, D. and Sansuc, J.-J., Descente et principe de Hasse pour certaines variétés rationnelles, J. Reine Angew. Math. 320 ( 1980), 150–191.Google Scholar
Colliot-Thélène, J.-L., Sansuc, J. and Swinnerton-Dyer, P., Intersection of two quadrics and Châtelet surfaces I, J. Reine Angew. Math. 373 (1987), 37–107.Google Scholar
Davenport, H. and Lewis, D. J., Non-homogeneous cubic equations, J. Lond. Math. Soc. 39 (1964), 657–671.CrossRefGoogle Scholar
Deligne, P., La conjecture de Weil I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.CrossRefGoogle Scholar
Demyanov, V. B., Pairs of quadratic forms over a complete field with discrete norm with a finite field of residue classes, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 307–324.Google Scholar
Fulton, W., Intersection Theory, 2nd ed. (Springer-Verlag, 1998).Google Scholar
Harris, J., Algebraic Geometry (GTM, Volume 133) (Springer-Verlag, 1992).Google Scholar
Hartshorne, R., Algebraic Geometry (GTM, Volume 52) (Springer-Verlag, 1977).Google Scholar
Heath-Brown, D. R., Cubic forms in ten variables, Proc. Lond. Math. Soc. 47 (1983), 225–257.CrossRefGoogle Scholar
Heath-Brown, D. R., A multiple exponential sum to modulus p 2, Canad. Math. Bull. 28 (1985), 394–396.Google Scholar
Heath-Brown, D. R., A new form of the circle method and its application to quadratic forms, J. Reine Angew. Math. 481 (1996), 149–206.Google Scholar
Heath-Brown, D. R., Cubic forms in 14 variables, Invent. Math. 170 (2007), 199–230.Google Scholar
Leep, D., Systems of quadratic forms, J. Reine Angew. Math. 350 (1984), 109–116.Google Scholar
Lewis, D. J., Cubic homogeneous polynomials over p-adic number fields, Ann. of Math. 56 (1952), 473–478.Google Scholar
Mordell, L. J., A remark on indeterminate equations in several variables, J. Lond. Math. Soc. 12 (1937), 127–129.Google Scholar
Pleasants, P. A. B., Cubic polynomials over algebraic number fields, J. Number Theory 7 (1975), 310–344.Google Scholar
Schmidt, W., The density of integer points on homogeneous varieties, Acta Math. 154 (1985), 243–296.Google Scholar
Swarbrick Jones, M., Weak approximation for cubic hypersurfaces of large dimension, Algebra & Number Theory 7 (2013), 1353–1363.CrossRefGoogle Scholar
Wooley, T. D., On simultaneous additive equations, II, J. Reine Angew. Math. 419 (1991), 141–198.Google Scholar
Wooley, T. D., On simultaneous additive equations, IV, Mathematika 45 (1998), 319–335.CrossRefGoogle Scholar
Zahid, J., Simultaneous zeros of a cubic and quadratic form, J. Lond. Math. Soc. 84 (2011), 612–630.CrossRefGoogle Scholar