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Zeros of systems of 𝔭-adic quadratic forms

Part of: Diophantine equations Forms and linear algebraic groups

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

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