Table of Contents
Preface viii
Part I Riemann-Roch Spaces
1 Elements of Algebra 2
1.1 Domains 2
1.2 Rings and Fields 9
1.3 Norm and Trace 13
1.4 Group of Units 15
2 Function Fields and Curves 17
2.1 Algebraic Function Fields 17
2.2 Affine Curves 19
2.3 Projective Curves 23
2.4 Genus of a Curve 27
3 Riemann-Roch Spaces 32
3.1 Valuation Rings 32
3.2 Places and Divisors 36
3.3 Riemann-Roch Space 39
3.4 Holomorphy Rings 41
4 Integral Domains 44
4.1 Fractional Ideals 44
4.2 Localization 46
4.3 R-module V 50
Part II Computation
5 Computing Integral Bases 54
5.1 Discriminant 54
5.2 Ideal Basis 56
5.3 Idealizer 58
5.4 Radical 60
5.5 Computing the Discriminants Ideal Radical 61
5.6 Computing Idealizers and Inverse Ideals 64
5.7 Trager's Algorithm 66
5.8 Examples 67
6 Computing Riemann-Roch Spaces 73
6.1 Reduced Basis Algorithms 73
6.2 The Dimension of L(D) 79
6.3 Computing L(D) 81
7 Computing Resultant and Norm Form Equations 85
7.1 Resultant 85
7.2 Computing NL/K(X1a1+…+Xnan) = K 90
7.3 Examples 92
8 Computing Integral Points on Rational Curves 98
8.1 Integral Points on Curves 98
8.2 Integral Points on Curves with |σ∞(C)| ≥ 3 99
8.3 Integral Points on Curves with |σ∞(C)| = 2 (i) 106
8.4 Integral Points on Curves with |σ∞(C)| = 2 (ii) 109
8.5 Integral Points on Curves with |σ∞(C)| = 1 112
Appendices 115
A Unimodular Matrices 115
A.1 Transformation Matrix 115
A.2 Unimodular Transformations 116
B Algorithm's Codes 118
B.1 Integral Points 3 Algorithm's Code 118
B.2 Integral Points 2i Algorithm's Code 126
B.3 Integral Points 2ii Algorithm's Code 129
B.4 Integral Points 1 Algorithm's Code 133
Bibliography 136
Index 139
Index of Algorithms 141