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Rational points on singular intersections of quadrics

Part of: Surfaces and higher-dimensional varieties Additive number theory; partitions Forms and linear algebraic groups Arithmetic problems. Diophantine geometry Diophantine equations

Published online by Cambridge University Press:  20 June 2013

T. D. Browning  and
R. Munshi
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK email [email protected]
R. Munshi
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai 400005, India email [email protected]
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Abstract

Given an intersection of two quadrics $X\subset { \mathbb{P} }^{m- 1} $, with $m\geq 9$, the quantitative arithmetic of the set $X( \mathbb{Q} )$ is investigated under the assumption that the singular locus of $X$ consists of a pair of conjugate singular points defined over $ \mathbb{Q} (i)$.

Keywords

Hardy–Littlewood circle methodpairs of quadratic formsrational points

MSC classification

Secondary: 11D72: Equations in many variables 11E12: Quadratic forms over global rings and fields 11P55: Applications of the Hardy-Littlewood method 14G25: Global ground fields 14J20: Arithmetic ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1457 - 1494
Copyright
© The Author(s) 2013 

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References

Aznar, V. N., On the Chern classes and the Euler characteristic for nonsingular complete intersections, Proc. Amer. Math. Soc. 78 (1980), 143–148.CrossRefGoogle Scholar
Birch, B. J., Forms in many variables, Proc. R. Soc. Lond. Ser. A 265 (1962), 245–263.Google Scholar
Browning, T. D., Heath-Brown, D. R. and Salberger, P., Counting rational points on algebraic varieties, Duke Math. J. 132 (2006), 545–578.Google Scholar
Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, P., Intersections of two quadrics and Châtelet surfaces, I, J. Reine Angew. Math. 373 (1987a), 37–107.Google Scholar
Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, P., Intersections of two quadrics and Châtelet surfaces, II, J. Reine Angew. Math. 374 (1987b), 72–168.Google Scholar
Cook, R. J., Simultaneous quadratic equations, J. Lond. Math. Soc. (2) 4 (1971), 319–326.Google Scholar
Davenport, H., Cubic forms in 16 variables, Proc. R. Soc. Lond. Ser. A 272 (1963), 285–303.Google Scholar
Deligne, P., La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.Google Scholar
Dem’yanov, 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 (in Russian).Google Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., Bounds for automorphic $L$-functions, Invent. Math. 112 (1993), 1–8.Google Scholar
Ein, L., Varieties with small dual varieties, I, Invent. Math. 86 (1986), 63–74.Google Scholar
Gelfand, I., Kapranov, M. and Zelevinsky, A., Discriminants, resultants, and multidimensional determinants (Birkhäuser, 1994).Google Scholar
Heath-Brown, D. R., Hybrid bounds for Dirichlet $L$-functions, II, Q. J. Math. 31 (1980), 157–167.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., Linear relations amongst sums of two squares, in Number theory and algebraic geometry, London Mathematical Society Lecture Note Series, vol. 303 (Cambridge University Press, Cambridge, 2003), 133–176.Google Scholar
Hooley, C., On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 32–98.Google Scholar
Hooley, C., On the number of points on a complete intersection over a finite field, J. Number Theory 38 (1991), 338–358.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Soceity. Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Iwaniec, H. and Munshi, R., The circle method and pairs of quadratic forms, J. Théor. Nombres Bordeaux 22 (2010), 403–419.Google Scholar
Loxton, J. H., Estimates for complete multiple exponential sums, Acta Arith. 92 (2000), 277–290.CrossRefGoogle Scholar
Reid, M., The complete intersection of two or more quadrics. PhD thesis, University of Cambridge (1972).Google Scholar
Smith, H. J. S., Arithmetical notes, Proc. Lond. Math. Soc. 4 (1873), 236–253.Google Scholar