## Algebraic Curves Over Finite FieldsIn 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

II | 1 |

III | 4 |

IV | 6 |

V | 8 |

VI | 10 |

VII | 11 |

VIII | 16 |

IX | 17 |

XLVI | 137 |

XLVII | 138 |

XLVIII | 140 |

L | 141 |

LI | 147 |

LII | 153 |

LIII | 154 |

LIV | 155 |

X | 18 |

XI | 20 |

XII | 23 |

XIII | 26 |

XIV | 27 |

XV | 28 |

XVI | 31 |

XVII | 32 |

XVIII | 36 |

XIX | 38 |

XX | 42 |

XXI | 46 |

XXII | 47 |

XXIII | 48 |

XXIV | 52 |

XXV | 57 |

XXVI | 59 |

XXVII | 69 |

XXVIII | 70 |

XXIX | 76 |

XXX | 85 |

XXXI | 87 |

XXXII | 89 |

XXXIII | 91 |

XXXIV | 92 |

XXXV | 95 |

XXXVI | 97 |

XXXVII | 98 |

XXXVIII | 108 |

XXXIX | 111 |

XL | 113 |

XLI | 114 |

XLII | 122 |

XLIII | 127 |

XLIV | 131 |

XLV | 136 |

LV | 160 |

LVI | 162 |

LVII | 164 |

LVIII | 170 |

LIX | 171 |

LX | 176 |

LXI | 178 |

LXII | 179 |

LXIII | 184 |

LXIV | 188 |

LXV | 189 |

LXVI | 198 |

LXVII | 203 |

LXVIII | 208 |

LXIX | 210 |

LXX | 211 |

LXXI | 212 |

LXXII | 213 |

LXXIII | 214 |

LXXIV | 216 |

LXXV | 217 |

LXXVI | 220 |

LXXVII | 221 |

LXXVIII | 222 |

LXXIX | 223 |

LXXX | 226 |

LXXXI | 227 |

LXXXII | 230 |

LXXXIII | 231 |

LXXXIV | 233 |

LXXXV | 234 |

LXXXVI | 235 |

239 | |

245 | |

### Other editions - View all

### Common terms and phrases

algebraic curves algebraic set algebraically closed automorphism character closed points closure coefficients complex numbers coordinate ring corresponding Definition degree denote differential dimension discrete valuation ring Div(C divisor classes element elliptic curve equation equivalent exceptional line extension fact fc[x field of constants finite field finite number formula function field Galois genus geometric given Goppa codes hence implies integral closure irreducible polynomials L-functions Lemma linear maximal ideal modular curves modulo monic irreducible polynomials morphism multiplicity non-zero notation number of closed number of points obtain ordp ordp(D point at infinity poles positive divisors positive integer pre-adele prime ideal principal divisor projective line projective line P1 projective plane curve proof properties proves quotient rational function regular functions Remark residue class field result Riemann hypothesis Riemann-Roch theorem satisfies Section singularities tangents uniformizing parameter variety vector space zeta function