Witt groups of conics, elliptic, and hyperelliptic curves
It is known that the Witt group of a real affine curve is finitely generated. The method of proof relies heavily on the fact that quadratic forms over function fields of real curves are determined by their classical invariants. Such a proof does not generalise (even for p-adic fields) to answer the following question: Let k be a field such that for every finite extension \(k'/k\), \(W(k')\) is finitely generated. Is the Witt group of any curve over k finitely generated? (The assumption ``W(k) finitely generated'' is not sufficient, in general, to ensure finite generation of Witt groups of curves over k. We prove section 3) that this is indeed true if the function field K(C) of the curve is contained in the quadratic closure of the rational function field. In particular, if C is a conic elliptic or a hyperelliptic curve over k, W(C) is finitely generated. The method of proof is to estimate the sizes of the kernel and cokernel of certain residue homomorphisms (see sections 2 and 3). We in fact determine precisely the kernel and cokernel of the residue homomorphisms for anisotropic conics (section 5). We use this to explicitly compute the Witt group of a conic over any field k of characteristic \(\neq 2\).
finite generation of Witt groups of curves
scientific article; zbMATH DE number 4139925