The local theory of root numbers.
Item
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Title
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The local theory of root numbers.
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Identifier
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AAI3115301
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identifier
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3115301
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Creator
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Wan, Aaron.
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Contributor
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Adviser: Carlos J. Moreno
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Date
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2004
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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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R. Langlands wrote his Yale manuscript on Arvin L-functions in 1971. In it he completes the proof of the existence and uniqueness of the local root number associated to each representation of a local Galois group. His original argument was strictly local, but P. Deligne later discovered a global analytic proof. By applying a number of Deligne's ideas, this thesis aims at simplifying Langlands' original local proof. The main tool is a variation of Brauer's induction theorem which decomposes every virtual representation of degree zero into a direct sum of induced representations of the form Ind(alpha - 1) where alpha - 1 denotes the difference between a one-dimensional character and the trivial character.;Central to Langlands' original argument are three fundamental root number identities. For tamely ramified field extensions, a coherent proof of these identities is presented in this thesis, using a method developed by B. Dwork who had earlier obtained two of the three root number identities.;While Langlands had originally established the local theory of root numbers in the context of Weil groups, J. Tate demonstrated that it is possible to formulate and solve the problem of existence and uniqueness without any reference to Weil groups if one takes Deligne's global approach. This thesis carries out Tate's viewpoint in a local context to as far as the nilpotent case and clarifies the role of Weil groups in Langlands' local approach for the general solvable case.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
