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Zeros of systems of 𝔭-adic quadratic forms

Part of: Forms and linear algebraic groups Diophantine equations

Published online by Cambridge University Press:  22 January 2010

D. R. Heath-Brown
D. R. Heath-Brown*
Affiliation:
Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB, UK (email: rhb@maths.ox.ac.uk)
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Abstract

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We show that a system of r quadratic forms over a 𝔭-adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax–Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.

Keywords

quadratic formsystem𝔭-adic fieldzerosAx–Kochen theorem

MSC classification

Secondary: 11E08: Quadratic forms over local rings and fields 11D72: Equations in many variables 11D79: Congruences in many variables 11D88: $p$-adic and power series fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 271 - 287
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Artin, E., The collected papers of Emil Artin (Addison-Wesley, Reading, MA, 1965).CrossRefGoogle Scholar
[2]Ax, J. and Kochen, S., Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630.Google Scholar
[3]Birch, B. J. and Lewis, D. J., Systems of three quadratic forms, Acta Arith. 10 (1964/1965), 423–442.Google Scholar
[4]Browning, T. D. and Heath-Brown, D. R., Counting rational points on hypersurfaces, J. Reine Angew. Math. 584 (2005), 83–115.CrossRefGoogle Scholar
[5]Davenport, H., Cubic forms in sixteen variables, Proc. Roy. Soc. Ser. A 272 (1963), 285–303.Google Scholar
[6]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
[7]Hasse, H., Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper, J. Reine Angew. Math. 153 (1924), 11–130.Google Scholar
[8]Heath-Brown, D. R., Zeros of p-adic forms, Proc. London Math. Soc. (3) (2009); doi: 10.1112/plms/pdp043.CrossRefGoogle Scholar
[9]Kneser, M., Quadratische formen (Springer, Berlin, 2002).CrossRefGoogle Scholar
[10]Leep, D. B., Systems of quadratic forms, J. Reine Angew. Math. 350 (1984), 109–116.Google Scholar
[11]Leep, D. B. and Schueller, L. M., A characterization of nonsingular pairs of quadratic forms, J. Algebr. Appl. 1 (2002), 391–412.CrossRefGoogle Scholar
[12]Martin, G., Solubility of systems of quadratic forms, Bull. London Math. Soc. 29 (1997), 385–388.CrossRefGoogle Scholar
[13]Schuur, S. E., On systems of three quadratic forms, Acta Arith. 36 (1980), 315–322.CrossRefGoogle Scholar
[14]Terjanian, G., Un contre-exemple à une conjecture d’Artin, C. R. Acad. Sci. Paris Sér. A–B 262 (1966), A612.Google Scholar