Nonstandard quadratic forms over rational function fields

Abstract

Let L=F(t1,…,tm) be a rational function field of characteristic different from 2. For a discrete F-valuation v on L denote by Lv the completion of L with respect to v.Assume φ is an anisotropic form over L, and let {ψ1,…,ψn} be a finite collection of anisotropic forms over F. We say that φ is stable with respect to {ψ1,…,ψn} if φK(t1,…,tm) is anisotropic for any extension K/F such that all the forms ψiK are anisotropic. The form φ is called nonstandard for the extension L/F if it is stable with respect to some collection of forms {ψ1,…,ψn}, and in addition for any discrete F-valuation v on L the form φLv is isotropic.Let X be a d-dimensional variety over an algebraically closed field k. We conjecture that if d≥1 and m≥1, then there is a nonstandard form φ with dim⁡φ≥2m+d−1+1 for the extension k(X)(t1,…,tm)/k(X). We prove this conjecture in the cases (d=2,m=1), (d=3,m=1), and (d=1,m=2). These cases are treated quite separately, by using different tools.In the last section we consider similar questions for systems of two quadratic forms.