Exponential sums and L-functions over finite fields.
Item
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Title
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Exponential sums and L-functions over finite fields.
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Identifier
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AAI9807911
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identifier
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9807911
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Creator
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Castro, Francis.
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Contributor
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Adviser: Carlos J. Moreno
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Date
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1997
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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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In this thesis we study several problems related to algebraic curves over finite fields and exponential sums in one and several variables. The following three topics are discussed:;(A) In Chapter 1 we make an analysis of the ramification groups of the composite field of an Artin-Schreier extension and a cyclic extension of degree {dollar}p\sp{lcub}i{rcub}{dollar} for {dollar}i=1,2.{dollar} This provides an effective method to calculate the conductor of the L-functions associated to such an extension. As a consequence of this calculation we are able to estimate certain mixed exponential sums constructed by multiplying an additive and a multiplicative character.;(B) In Chapter 2 we study algebraic curves with singularities and introduce a new definition of L-function associated to an abelian covering of the projective line. The main result we prove shows that our definition is compatible with that of the zeta function defined by Stohr. We have calculated several examples and one is particularly interesting because it shows that the L-function may not be a rational function. The main application we give of these results is to the definition of exponential sums over singular curves. We give estimates for such sums.;(C) Chapter 3 deals mainly with exponential sums in several variables. We make a detailed study of the Kloosterman sum in seven variables. For this sum we determine its L-function. As an application of this result we obtain a Weyl-type distribution for the signs of the Kloosterman sum.;We have included an introductory section (Chapter 0) that serves to set down the notation and basic definitions in the classical theory of exponential sums.
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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.
