On the U-Invariant of P-Adic Function Fields

ABSTRACT

We show that a quadratic form defined over the rational function field ℚ(x ₁ , …, _{x n} ) of dimension at least 4.2 ⁿ  + 1 is isotropic over all fields ℚ _p (x ₁ , …, _{x n} ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.

Key Words:

• Function fields
• Model theory
• Quadratic forms

Mathematics Subject Classification:

• Primary 03C60
• 12L12
• Secondary 12F20
• 12E30

ACKNOWLEDGMENTS

This paper was written during the author's stay at the Equipe de Logique Mathématique at the Université de Paris 7. During his stay, the author was supported through a Marie Curie Individual Fellowship (Contract HPMF-CT-2001-01384). Part of the work was done while the author was staying at the Department of Mathematics at University College Dublin, during which the author was holding a post-doctoral position in the TMR network “Linear algebraic groups, K-theory and related topics”.

Notes

¹A similar statement also follows from a theorem by Greenleaf (1965).

^#Communicated by H. Macpherson.