The Witt Ring of a Smooth Curve with Good Reduction over a Local Field
Item
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Title
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The Witt Ring of a Smooth Curve with Good Reduction over a Local Field
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Identifier
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d_2009_2013:a5e531a8bf7a:11438
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identifier
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11850
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Creator
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Funk, Jeanne M.,
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Contributor
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Raymond T. Hoobler
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Date
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2012
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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The modern study of bilinear forms has a rich history beginning with Witt's work over fields in the 1930's, when he defined a ring structure on the set of anisotropic forms over a field. It was revived, notably by Pfister, in the 1960's. With the advent of algebraic K-Theory, much of the theory of quadratic forms over fields was generalized to a theory of quadratic spaces over rings. In the 1960's and 1970's Knebusch, among others, formulated a compatible theory for quadratic forms over schemes in which a ring analogous to Witt's ring of anisotropic forms is prominent. Calculation of such "Witt rings" is a problem of interest in modern algebraic geometry.;This thesis focuses on the calculation of the Witt ring of a smooth geometrically connected curve with good reduction over a local field. As a sub-problem, we calculate the Witt ring of a smooth geometrically connected curve over a finite field. We present a generalization to the category of sheaves of the filtration of the Witt ring by powers of its fundamental ideal of even rank elements. This yields a filtration by global sections which we study using etale cohomology. In the cases of interest here, this allows us to describe the Witt classes of a curve in terms of the classical invarients rank, signed discriminant, and Witt invariant.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
