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RATIONAL POINTS ON INTERSECTIONS OF CUBIC AND QUADRIC HYPERSURFACES
Part of:
Additive number theory; partitions
Arithmetic problems. Diophantine geometry
Arithmetic algebraic geometry
Published online by Cambridge University Press: 05 June 2014
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
MSC classification
- 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
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
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
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
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
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