By: Carlos Moreno, B. Bollobas (Editor), W. Fulton (Editor), A. Katok (Editor), F. Kirwan (Editor)

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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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Preface	p. ix
Algebraic curves and function fields	p. 1
Geometric aspects	p. 1
Introduction	p. 1
Affine varieties	p. 1
Projective varieties	p. 4
Morphisms	p. 6
Rational maps	p. 8
Non-singular varieties	p. 10
Smooth models of algebraic curves	p. 11
Algebraic aspects	p. 16
Introduction	p. 16
Points on the projective line P[superscript 1]	p. 17
Extensions of valuation rings	p. 18
Points on a smooth curve	p. 20
Independence of valuations	p. 23
Exercises	p. 26
Notes	p. 27
The Riemann-Roch theorem	p. 28
Divisors	p. 28
The vector space L(D)	p. 31
Principal divisors and the group of divisor classes	p. 32
The Riemann theorem	p. 36
Pre-adeles (repartitions)	p. 38
Pseudo-differentials (the Riemann-Roch theorem)	p. 42
Exercises	p. 46
Notes	p. 47
Zeta functions	p. 48
Introduction	p. 48
The zeta functions of curves	p. 48
The functional equation	p. 52
Consequences of the functional equation	p. 57
The Riemann hypothesis	p. 59
The L-functions of curves and their functional equations	p. 69
Preliminary remarks and notation	p. 69
Algebraic aspects	p. 70
Geometric aspects	p. 76
Exercises	p. 85
Notes	p. 87
Exponential sums	p. 89
The zeta function of the projective line	p. 89
Gauss sums: first example of an L-function for the projective line	p. 91
Properties of Gauss sums	p. 92
Cyclotomic extensions: basic facts	p. 92
Elementary properties	p. 95
The Hasse-Davenport relation	p. 97
Stickelberger's theorem	p. 98
Kloosterman sums	p. 108
Second example of an L-function for the projective line	p. 108
A Hasse-Davenport relation for Kloosterman sums	p. 111
Third example of an L-function for the projective line	p. 113
Basic arithmetic theory of exponential sums	p. 114
Part I: L-functions for the projective line	p. 114
Part II: Artin-Schreier coverings	p. 122
The Hurwitz-Zeuthen formula for the covering [pi]: C [right arrow] C	p. 127
Exercises	p. 131
Notes	p. 136
Goppa codes and modular curves	p. 137
Elementary Goppa codes	p. 138
The affine and projective lines	p. 140
Affine line A[superscript 1](k)	p. 140
Projective line P[superscript 1]	p. 141
Goppa codes on the projective line	p. 147
Algebraic curves	p. 153
Separable extensions	p. 154
Closed points and their neighborhoods	p. 155
Differentials	p. 160
Divisors	p. 162
The theorems of Riemann-Roch, of Hurwitz and of the Residue	p. 164
Linear series	p. 170
Algebraic geometric codes	p. 171
Algebraic Goppa codes	p. 171
Codes with better rates than the Varshamov-Gilbert bound	p. 176
The theorem of Tsfasman, Vladut and Zink	p. 178
Modular curves	p. 178
Elliptic curves over C	p. 179
Elliptic curves over the fields F[subscript p], Q	p. 184
Torsion points on elliptic curves	p. 188
Igusa's theorem	p. 189
The modular equation	p. 198
The congruence formula	p. 203
The Eichler-Selberg trace formula	p. 208
Proof of the theorem of Tsfasman, Vladut and Zink	p. 210
Examples of algebraic Goppa codes	p. 211
The Hamming (7,4) code	p. 212
BCH codes	p. 213
The Fermat cubic (Hermite form)	p. 214
Elliptic codes (according to Driencourt-Michon)	p. 216
The Klein quartic	p. 217
Exercises	p. 220
Simplification of the singularities of algebraic curves	p. 221
Homogeneous coordinates in the plane	p. 222
Basic lemmas	p. 223
Dual curves	p. 226
Plucker formulas	p. 227
Quadratic transformations	p. 230
Quadratic transform of a plane curve	p. 231
Quadratic transform of a singularity	p. 233
Singularities off the exceptional lines	p. 234
Reduction of singularities	p. 235
Bibliography	p. 239
Index	p. 245
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521459013
ISBN-10: 052145901X
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 260
Published: 14th October 1993
Publisher: Cambridge University Press
Dimensions (cm): 22.8 x 15.2  x 1.5
Weight (kg): 0.39