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Pairs of quadrics in 11 variables

Part of: Diophantine equations

Published online by Cambridge University Press:  23 March 2015

Ritabrata Munshi
Ritabrata Munshi*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1, Dr Homi Bhabha Road, Colaba, Mumbai 400005, India email [email protected]
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Abstract

For non-singular intersections of pairs of quadrics in 11 or more variables, we prove an asymptotic for the number of rational points in an expanding box.

Keywords

circle methodquadratic formsrational points

MSC classification

Primary: 11D72: Equations in many variables
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 7 , July 2015 , pp. 1189 - 1214
Copyright
© The Author 2015 

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References

Birch, B. J., Forms in many variables, Proc. R. Soc. Lond. Ser. A 265 (1962), 245–263.Google Scholar
Birch, B. J., Lewis, D. J. and Murphy, T. G., Simultaneous quadratic forms, Amer. J. Math. 84 (1962), 110–115.CrossRefGoogle Scholar
Browning, T. D., Dietmann, R. and Heath-Brown, D. R., Rational points on intersections of cubic and quadric hypersurfaces, J. Inst. Math. Jussieu, to appear, arXiv:1309.0147.Google Scholar
Browning, T. D. and Munshi, R., Rational points on singular intersections of quadrics, Compositio Math. 149 (2013), 1457–1494.CrossRefGoogle Scholar
Browning, T. D. and Munshi, R., Pairs of diagonal quadratic forms and linear correlations among sums of two squares, Forum Math., to appear.Google Scholar
Cook, R. J., Simultaneous quadratic equations, J. Lond. Math. Soc. (2) 4 (1971), 319–326.CrossRefGoogle 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 (1987), 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 (1987), 72–168.Google Scholar
Davenport, H., Cubic forms in 32 variables, Philos. Trans. R. Soc. Lond. A 251 (1959), 193–232.Google 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
Dietmann, R., Systems of rational quadratic forms, Arch. Math. 82 (2004), 507–516.CrossRefGoogle Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., Bounds for automorphic L-functions, Invent. Math. 112 (1993), 1–8.CrossRefGoogle Scholar
Heath-Brown, D. R., A new form of the circle method, and its applications 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.CrossRefGoogle Scholar
Heath-Brown, D. R., Zeros of systems of p-adic quadratic forms, Compositio Math. 146 (2010), 271–287.CrossRefGoogle Scholar
Heath-Brown, D. R., Zeros of pairs of quadratic forms, Preprint (2013), arXiv:1304.3894.Google Scholar
Heath-Brown, D. R. and Pierce, L., Simultaneous integer values of pairs of quadratic forms, J. Reine Angew. Math., to appear, arXiv:1309.6767.Google Scholar
Reid, M., The complete intersection of two or more quadrics, PhD thesis, Cambridge (1972).Google Scholar
Schmidt, W., Simultaneous rational zeros of quadratic forms, in Seminar on number theory, Paris 1980–1981, Progress in Mathematics vol. 22 (Birkhäuser, Boston, MA, 1982), 281–307.Google Scholar