## An Invitation to Arithmetic Geometry |

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### Contents

Integral closure | 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