Zeros of systems of p-adic quadratic forms

Author: 

Heath-Brown, D

Publication Date: 

1 March 2010

Journal: 

Compositio Mathematica

Last Updated: 

2019-05-02T16:26:06.86+01:00

Issue: 

2

Volume: 

146

DOI: 

10.1112/S0010437X09004497

page: 

271-287

abstract: 

We show that a system of r quadratic forms over a -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. © 2010 Foundation Compositio Mathematica.

Symplectic id: 

53389

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article