On 1-isometries of affine quadrics over finite fields

Abstract

Letq be a regular quadratic form on a vector space (V,\(\mathbb{F}\)) and assume\(4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}\). A 1-isometry of the central quadric\(\mathcal{F}: = \{ x \in V|q(x) = 1\}\) is a permutation ϕ of\(\mathcal{F}\) such that $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ (*) holds true for a fixed element ν of\(\mathbb{F}\). For arbitraryν ∈\(\mathbb{F}\) we prove thatϕ is induced (in a certain sense) by a semi-linear bijection\((\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})\) such thatq oσ =ϱ oq, provided\(\mathcal{F}\) contains lines and the exceptional case\((\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)\) is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.