## An Invitation to Arithmetic GeometryExtremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature. --Zentralblatt MATH The interaction between number theory and algebraic geometry has been especially fruitful. In this volume, the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes, which will aid the reader who goes to the next level of this rich subject. |

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### Contents

5 | |

Plane curves | 35 |

Factorization of ideals | 85 |

The discriminants | 131 |

The ideal class group | 157 |

Projective curves | 193 |

Nonsingular complete curves | 225 |

Zetafunctions | 269 |

The RiemannRoch Theorem | 305 |

Frobenius morphisms and the Riemann hypothesis | 339 |

Further topics | 361 |

Appendix | 375 |

### Common terms and phrases

absolute value affine curve algebraically closed automorphism bijection Chapter char(fc closure of k[x coefficients concludes the proof contains Corollary curve Z/(k Dedekind domain defined definition deg(D denote the integral dimension Div(X divisor domain with field element equal Example exer exists factorization of ideals fc-algebras field extension field of fractions finite extension finite field finitely generated A-module follows Frobenius function field Galois extension Galois group genus geometry Hence homogeneous polynomial injective integral closure irreducible polynomial isomorphic K.Let Lemma Let B denote Let f Let us assume Let X/k Max(A Max(C maximal ideal minimal polynomial monic morphism of curves noetherian non-zero nonsingular complete curve nonsingular curve number fields plane curve plane projective curve polynomial of degree prime ideal principal ideal domain Proposition prove quotient ramified Riemann hypothesis Riemann-Roch Theorem ring of functions S~lA shows squarefree subfield subgroup subring surjective unramified valuation XF(k Xp(k zeta-function Zg(k

### Popular passages

Page 25 - Let 0 — * M' — * M —^-* M" — . 0 be an exact sequence of A-modules. Then M is noetherian if and only if M' and M

Page 20 - Recall that an ideal / in a ring A is said to be finitely generated if there exist finitely many elements GI , . . . , cr in / such that / = {aici H i- arcr \ tn € A, i = 1, . . . , r).

Page 16 - Let B be the integral closure of A in L. Then B is finitely generated as a K -algebra, hence Noetherian.