Nonstandard quadratic forms over rational function fields

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 cp is an anisotropic form over L, and let {psi 1, ... , psi n} be a finite collection of anisotropic forms over F. We say that cp is stable with respect to {psi 1, ..., psi n} if cpK(t1,...,tm) is anisotropic for any extension K/F such that all the forms psi iK are anisotropic. The form cp is called nonstandard for the extension L/F if it is stable with respect to some collection of forms {psi 1, . .. , psi n}, and in addition for any discrete F-valuation v on L the form cpLv 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 cp with dim cp > 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.(c) 2022 Elsevier B.V. All rights reserved.