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Algebraic Curves over Finite Fields : Cambridge Tracts in Mathematics (Paperback) - Carlos Moreno

Algebraic Curves over Finite Fields

Cambridge Tracts in Mathematics (Paperback)

Paperback Published: 13th December 1993
ISBN: 9780521459013
Number Of Pages: 260

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In this tract, Professor Moreno develops the theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves. Among the applications considered are: the problem of counting the number of solutions of equations over finite fields; Bombieri's proof of the Reimann hypothesis for function fields, with consequences for the estimation of exponential sums in one variable; Goppa's theory of error-correcting codes constructed from linear systems on algebraic curves; there is also a new proof of the TsfasmanSHVladutSHZink theorem. The prerequisites needed to follow this book are few, and it can be used for graduate courses for mathematics students. Electrical engineers who need to understand the modern developments in the theory of error-correcting codes will also benefit from studying this work.

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Prefacep. ix
Algebraic curves and function fieldsp. 1
Geometric aspectsp. 1
Introductionp. 1
Affine varietiesp. 1
Projective varietiesp. 4
Morphismsp. 6
Rational mapsp. 8
Non-singular varietiesp. 10
Smooth models of algebraic curvesp. 11
Algebraic aspectsp. 16
Introductionp. 16
Points on the projective line P[superscript 1]p. 17
Extensions of valuation ringsp. 18
Points on a smooth curvep. 20
Independence of valuationsp. 23
Exercisesp. 26
Notesp. 27
The Riemann-Roch theoremp. 28
Divisorsp. 28
The vector space L(D)p. 31
Principal divisors and the group of divisor classesp. 32
The Riemann theoremp. 36
Pre-adeles (repartitions)p. 38
Pseudo-differentials (the Riemann-Roch theorem)p. 42
Exercisesp. 46
Notesp. 47
Zeta functionsp. 48
Introductionp. 48
The zeta functions of curvesp. 48
The functional equationp. 52
Consequences of the functional equationp. 57
The Riemann hypothesisp. 59
The L-functions of curves and their functional equationsp. 69
Preliminary remarks and notationp. 69
Algebraic aspectsp. 70
Geometric aspectsp. 76
Exercisesp. 85
Notesp. 87
Exponential sumsp. 89
The zeta function of the projective linep. 89
Gauss sums: first example of an L-function for the projective linep. 91
Properties of Gauss sumsp. 92
Cyclotomic extensions: basic factsp. 92
Elementary propertiesp. 95
The Hasse-Davenport relationp. 97
Stickelberger's theoremp. 98
Kloosterman sumsp. 108
Second example of an L-function for the projective linep. 108
A Hasse-Davenport relation for Kloosterman sumsp. 111
Third example of an L-function for the projective linep. 113
Basic arithmetic theory of exponential sumsp. 114
Part I: L-functions for the projective linep. 114
Part II: Artin-Schreier coveringsp. 122
The Hurwitz-Zeuthen formula for the covering [pi]: C [right arrow] Cp. 127
Exercisesp. 131
Notesp. 136
Goppa codes and modular curvesp. 137
Elementary Goppa codesp. 138
The affine and projective linesp. 140
Affine line A[superscript 1](k)p. 140
Projective line P[superscript 1]p. 141
Goppa codes on the projective linep. 147
Algebraic curvesp. 153
Separable extensionsp. 154
Closed points and their neighborhoodsp. 155
Differentialsp. 160
Divisorsp. 162
The theorems of Riemann-Roch, of Hurwitz and of the Residuep. 164
Linear seriesp. 170
Algebraic geometric codesp. 171
Algebraic Goppa codesp. 171
Codes with better rates than the Varshamov-Gilbert boundp. 176
The theorem of Tsfasman, Vladut and Zinkp. 178
Modular curvesp. 178
Elliptic curves over Cp. 179
Elliptic curves over the fields F[subscript p], Qp. 184
Torsion points on elliptic curvesp. 188
Igusa's theoremp. 189
The modular equationp. 198
The congruence formulap. 203
The Eichler-Selberg trace formulap. 208
Proof of the theorem of Tsfasman, Vladut and Zinkp. 210
Examples of algebraic Goppa codesp. 211
The Hamming (7,4) codep. 212
BCH codesp. 213
The Fermat cubic (Hermite form)p. 214
Elliptic codes (according to Driencourt-Michon)p. 216
The Klein quarticp. 217
Exercisesp. 220
Simplification of the singularities of algebraic curvesp. 221
Homogeneous coordinates in the planep. 222
Basic lemmasp. 223
Dual curvesp. 226
Plucker formulasp. 227
Quadratic transformationsp. 230
Quadratic transform of a plane curvep. 231
Quadratic transform of a singularityp. 233
Singularities off the exceptional linesp. 234
Reduction of singularitiesp. 235
Bibliographyp. 239
Indexp. 245
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521459013
ISBN-10: 052145901X
Series: Cambridge Tracts in Mathematics (Paperback)
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 260
Published: 13th December 1993
Country of Publication: GB
Dimensions (cm): 22.76 x 15.24  x 1.5
Weight (kg): 0.37

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