Hyperbolicity and near hyperbolicity of quadratic forms over function\n fields of quadrics

2016/09/22 by Stephen Scully, Scully, Stephen
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1609.07100

Abstract

Let p and q be anisotropic quadratic forms over a field F of\ncharacteristic \≠ 2, let s be the unique non-negative integer such that\n2^s < dim(p) \≤ 2^s+1, and let k denote the dimension of the\nanisotropic part of q after scalar extension to the function field F(p) of\np. We conjecture that dim(q) must lie within k of a multiple of\n2^s+1. This can be viewed as a direct generalization of Hoffmann's\nseparation theorem. Among other cases, we prove that the conjecture is true if\nk<2^s-1. When k=0, this shows that any anisotropic form representing an\nelement of the kernel of the natural restriction homomorphism W(F)\→\nW(F(p)) has dimension divisible by 2^s+1.\n

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