| 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 |
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