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Quadratic lattices in function fields of genus 0. (English) Zbl 0808.11025

Let \(K/k\) be a function field of genus 0, where \(k\) is a field of characteristic not 2, and let \(R\) be the ring of integers in \(K\). Quadratic form theory over \(K\) and \(R\) is well understood if \(K\) is the rational function field over \(k\). The author specifies four principal results supporting this statement. They include the results on successive minima of quadratic lattices over \(R\), and three exact sequences of Witt groups showing how \(W(k)\), \(W(R)\) and \(W(k)\) sit in the Witt groups \(W(R)\), \(W(K)\) and \(W(K)\), respectively. These four results are generalized here to the context of arbitrary function fields of genus 0. On the way the author systemizes the general theory and gives fresh proofs of some closely related results (such as Arason’s excellence theorem). The author’s aim is to use only “elementary” methods such as the classical methods of integral quadratic forms and the methods of algebraic theory of quadratic forms, excluding function field theory, \(K\)- and \(L\)-theory and algebraic geometry.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
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