Higher Genus Curves in Mathematical Physics and Arithmetic Geometry

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Cover......Page 1 Title page......Page 2 Contents......Page 4 Preface......Page 6 1. Introduction......Page 10 2. Preliminaries......Page 11 3. Bounding Actions by the Length of the Tail......Page 14 4. A Lower Bound for ��_{��_{��},��}......Page 18 References......Page 19 1. Introduction......Page 22 2. Quasi-platonic group actions and regular Belyi functions......Page 24 3. The Galois action on quasi-platonic actions......Page 30 4. Examples of Galois actions on quasi-platonic actions......Page 37 References......Page 41 Equations of Riemann surfaces with automorphisms......Page 42 1. The main algorithm......Page 43 2. Example: A genus 7 Riemann surface with 54 automorphisms......Page 47 3. Selected results......Page 49 References......Page 53 1. Introduction......Page 56 2. Preliminaries......Page 59 3. Field of moduli of superelliptic curves......Page 61 4. Superelliptic curves of genus at most 10......Page 63 5. Tables of superelliptic curves of genus between 5 and 10......Page 65 References......Page 70 1. Introduction......Page 72 2. Reduction of binary quintics and sextics......Page 74 3. Julia quadratic of genus two curves with extra automorphisms......Page 81 4. Minimal models of curves with extra involutions......Page 84 5. Some heuristics for curves with extra involutions defined over Q......Page 87 References......Page 91 1. Introduction......Page 92 2. A database of integral binary sextics......Page 94 3. Heights of genus two curves......Page 95 4. Genus 2 curves over C......Page 97 5. Algebraic invariants......Page 100 6. Automorphisms......Page 106 7. Genus 2 curves defined over Q......Page 109 8. Minimal discriminant for Weierstrass equations......Page 112 9. Constructing the databases......Page 115 Creating the databases......Page 119 Appendix B. Basic Invariants and relations among them......Page 120 References......Page 122 1. Introduction......Page 126 2. Hypersurfaces in toric varieties......Page 129 3. Elliptic curves......Page 131 4. Experimental evidence for strong mirror symmetry......Page 132 5. Picard-Fuchs equations......Page 134 References......Page 137 1. Introduction......Page 140 2. Preliminaries......Page 142 3. Inose type surface for the Jacobian of a curve of genus 2......Page 143 4. Fibration with two ����* fibers......Page 146 References......Page 149 1. Introduction......Page 152 2. Preliminaries and notation......Page 154 3. A basis of holomorphic q-differentials......Page 155 4. Weights of branch points......Page 158 References......Page 164 Introduction......Page 166 1. Limits and Invariants......Page 167 2. Monodromy of ��-gons......Page 174 References......Page 177 1. Introduction......Page 180 2. Notation and definitions......Page 181 3. Runge’s method......Page 183 5. Main theorem......Page 185 6. Relation to Runge’s method......Page 187 7. Algebraic curves......Page 188 8. Higher-dimensional varieties......Page 194 References......Page 196 1. Introduction......Page 198 2. Self-inversive polynomials......Page 199 3. Superelliptic curves and self-inversive polynomials......Page 207 4. Self-reciprocal polynomials and reduction theory......Page 210 5. Self-reciprocal polynomials and codes......Page 213 References......Page 216 1. Introduction......Page 218 2. Divisors on Hurwitz spaces......Page 220 3. The generic splitting type......Page 225 4. The divisor class of mu......Page 226 References......Page 230 Back Cover......Page 234