Algebraic curves and their applications

Cover......Page 1
Title page......Page 4
Contents......Page 6
Preface......Page 8
Families of elliptic curves with rational torsion points of even order......Page 12
1. Introduction......Page 12
2. Review of [1]......Page 14
3. Division by 2......Page 16
4. Another criterion of divisibility by 2......Page 21
5. Elliptic curves with points of order 4......Page 26
6. Elliptic curves with points of order 8......Page 28
7. Elliptic curves with point of order 6......Page 32
8. Elliptic curves with point of order 12......Page 35
9. Elliptic curves with rational points of order 10......Page 37
10. Elliptic curves in characteristic 2......Page 38
References......Page 43
The weighted moduli space of binary sextics......Page 44
1. Introduction......Page 44
2. Preliminaries......Page 46
3. Rational points in the weighted moduli space......Page 50
Acknowledgment......Page 54
References......Page 54
A family of nonnormal double planes associated to hyperelliptic curves......Page 56
1. Introduction......Page 56
2. Background material......Page 58
3. The main theorem......Page 60
4. Examples and applications......Page 62
References......Page 64
On the discriminant of certain quadrinomials......Page 66
1. Introduction......Page 66
2. Preliminaries......Page 67
3. Computation of the matrix .._{scriptsize\boldmath..}(..)_{..-2}......Page 69
4. Proof of theorems......Page 78
5. Concluding remarks......Page 80
References......Page 81
Semistable types of hyperelliptic curves......Page 84
1. Introduction......Page 84
2. Background on metric graphs......Page 89
3. Hyperelliptic graphs, BY trees and cluster pictures......Page 91
4. One-to-one correspondence (open case)......Page 101
5. One-to-one correspondence (closed case)......Page 115
6. The homology lattice .......Page 125
7. Tamagawa groups of hyperelliptic graphs......Page 131
8. Classification of semistable types and naming convention......Page 137
9. Classification for small genera......Page 141
Acknowledgements......Page 145
References......Page 145
Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function......Page 148
1. Introduction......Page 148
2. Representability results concerning Hom......Page 149
3. Projective systems of Hilbert spaces......Page 150
4. Finite free algebras......Page 151
5. Auto-arc spaces of formal deformations......Page 153
6. A generalization of the motivic Igusa zeta function......Page 154
7. Motivic rationality for plane curve singularities......Page 156
References......Page 157
Computing heights on weighted projective spaces......Page 160
1. Introduction......Page 160
2. Weight projective spaces......Page 162
3. Heights on the weighted projective spaces......Page 166
4. Computing the weighted height......Page 168
References......Page 169
On hyperelliptic curves of genus 3......Page 172
1. Introduction......Page 172
2. Preliminaries......Page 173
3. Weighted moduli space of binary octavics......Page 176
4. A database of genus 3 hyperelliptic curves......Page 179
5. Concluding remarks......Page 180
Acknowledgment......Page 180
References......Page 183
On automorphisms of algebraic curves......Page 186
1. Introduction......Page 186
2. Algebraic curves and their function fields......Page 188
3. Weierstrass Gap theorem and Weierstrass points......Page 194
4. Automorphisms of curves......Page 197
5. Superelliptic curves......Page 200
6. Automorphism groups of compact Riemann surfaces......Page 206
References......Page 220
On the algebraic classification of subgroups of hyperbolic planar crystallographic groups......Page 224
1. Introduction......Page 224
2. Canonical presentation......Page 227
......Page 227
4. Examples where direction of period-cycles is essential to know if two subgroups of a hyperbolic crystallographic group are isomorphic......Page 229
5. Computing the signature of subgroups using a package of GAP......Page 232
Acknowledgments......Page 233
References......Page 233
......Page 236
1. Introduction......Page 236
2. Background......Page 237
......Page 239
References......Page 243
An explicit descent of real algebraic varieties......Page 246
1. Introduction......Page 246
2. Main results......Page 247
3. Proof of Theorems 1 and 2......Page 249
4. An example......Page 253
5. MAGMA implementation......Page 255
Acknowledgments......Page 256
References......Page 256
Curves in isomonodromy and isospectral deformations: Painleve VI as a case study......Page 258
1. Introduction......Page 258
2. Connecting integrable PDEs and Painleve-type. I: Isomonodromy aspects......Page 261
3. Connecting integrable PDEs and Painleve-type. II: Symmetry reduction......Page 266
4. Comments and further aspects......Page 272
Acknowledgment......Page 273
References......Page 273
Quasi-quadratic residue codes and hyperelliptic curves......Page 278
1. Quasi-quadratic residue codes......Page 278
2. Hyperelliptic curves......Page 283
References......Page 289
Curves, Jacobians, and cryptography......Page 290
Preface......Page 290
Part 1. Abelian varieties......Page 293
1. Definitions and basic properties......Page 293
2. Endomorphisms and isogenies......Page 297
3. Projective curves and Jacobian varieties......Page 302
4. Applications of the Riemann-Roch theorem......Page 307
5. Modular curves......Page 325
Part 2. Cryptography......Page 328
6. Diffie-Hellman key exchange......Page 328
7. Index calculus in Picard groups......Page 332
8. Isogenies of Jacobians via correspondences and applications to discrete logarithms......Page 335
9. Genus 3 curves and cryptography......Page 338
10. Genus 2 curves and cryptography......Page 342
11. Elliptic curve cryptography......Page 348
References......Page 352
Back Cover......Page 358