Witt groups of conics, elliptic, and hyperelliptic curves
type: Item
instance of: scholarly article
title: Witt groups of conics, elliptic, and hyperelliptic curves
zbMATH Open document ID: 0696.14016
DOI: 10.1016/0022-314X(88)90120-5
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: 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 \(\neq 2\).
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 4139925
zbMATH Keywords: characteristic p
zbMATH Keywords: finite generation of Witt groups of curves
zbMATH Keywords: Witt group of a conic
author: R. Parimala
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4139925