The development of new computational techniques and better computing power has made it possible to attack some classical problems of algebraic geometry. The main goal of this book is to highlight such computational techniques related to algebraic curves. The area of research in algebraic curves is receiving more interest not only from the mathematics community, but also from engineers and computer scientists, because of the importance of algebraic curves in applications including cryptography, coding theory, error-correcting codes, digital imaging, computer vision, and many more. This book covers a wide variety of topics in the area, including elliptic curve cryptography, hyper elliptic curves, representations on some Riemann-Roch spaces of modular curves, computation of Hurwitz spectra, generating systems of finite groups, and Galois groups of polynomials, among other topics.
Author(s): Tanush Shaska
Series: Lecture notes series on computing 13
Publisher: World Scientific
Year: 2005
Language: English
Pages: 286
City: Hackensack, NJ
CONTENTS ......Page 6
1. Preface ......Page 8
2. Foreword by the Editor ......Page 10
1. Introduction ......Page 14
2. Overview of mathematical methods ......Page 15
3. Description of the first pairing ......Page 17
4. Proof of non-degeneracy of the first pairing ......Page 18
5. Motivation for the pairing using isogenies over the base field ......Page 20
6. Applications to elliptic curves ......Page 24
1. Introduction ......Page 26
2. Division Polynomials ......Page 28
3. E(Q)tors and Hensel's lemma ......Page 29
4. Deciding whether E(Qp)[p] is non-trivial ......Page 30
5. Computing E(Q)[2] ......Page 37
6. The p-adic algorithm ......Page 38
7. Discriminant of the division polynomial ......Page 41
8. The l-adic algorithm ......Page 44
9. Time complexity analysis of the algorithms ......Page 47
10. Conclusions ......Page 49
1. Introduction ......Page 51
2. Background ......Page 53
3. Algorithms for elliptic curves ......Page 55
4. Algorithmic considerations ......Page 61
1. Myrberg's algorithm for elliptic curves ......Page 64
2. Generalizations of Myrberg's algorithm ......Page 69
1. Introduction ......Page 71
2. Hurwitz Spaces ......Page 72
3. The geometry of Hg2 ......Page 75
4. The geometry of Hg3 ......Page 82
1. Preliminaries on genus 2 curves ......Page 84
2. The case Aut(C) - V4 ......Page 89
3. The case Aut(C)-D8 D12 ......Page 93
4. The isolated cases ......Page 95
1. Introduction ......Page 97
2. The combinatorial technique ......Page 98
3. Computing the traces ......Page 100
4. Structure of the trace sequence ......Page 102
1. Introduction ......Page 108
2. Fuchsian Groups and Uniformization ......Page 109
3. Finding Signatures ......Page 111
4. Finding Non-Normal Belyi Surfaces ......Page 115
1. Introduction ......Page 122
2. Dihedral invariants of hyperelliptic curves ......Page 124
3. Hyperelliptic curves of genus three ......Page 128
4. Field of moduli of genus 3 hyperelliptic curves ......Page 134
1. Introduction ......Page 137
2. Goppa codes ......Page 138
3. Upper bounds ......Page 140
4. Lower bounds ......Page 150
5. Determination of Nq(g) ......Page 152
1. Introduction ......Page 158
2. Preliminaries ......Page 159
3. Hyperelliptic curves of genus 3 ......Page 160
4. Hyperelliptic curves of genus 4 ......Page 166
5. APPENDIX ......Page 173
1. Introduction ......Page 176
2. Modular curves ......Page 177
3. Representation theory of PSL(2 N) ......Page 182
4. Induced characters ......Page 185
5. Ramification module ......Page 192
6. Equivariant degree and Riemann-Roch space ......Page 197
7. Examples ......Page 200
8. Application to codes ......Page 206
9. A G codes associated to X(7) ......Page 208
10. Appendix A: Tables for Theorem 9 ......Page 210
11. Appendix B: Ramification modules and GAP code ......Page 214
1. Introduction ......Page 219
2. Preliminaries ......Page 222
3. Curves of genus 2 with split Jacobians ......Page 223
4. The locus of genus two curves with (n n) split Jacobians ......Page 227
5. Genus 2 curves with degree 2 elliptic subcovers ......Page 229
6. Genus 2 curves with degree 3 elliptic subcovers ......Page 235
7. Further remarks ......Page 243
1. Introduction ......Page 245
2. Cohomology ......Page 246
3. Curves ......Page 248
4. Degree zero ......Page 249
5. Examples ......Page 251
1. Introduction ......Page 256
2. The Galois group of a polynomial ......Page 258
3. Polynomials with non-real roots ......Page 260
4. Concluding remarks ......Page 267
1. Introduction ......Page 269
3. Second step of the algorithm ......Page 271
4. An application: tuples of index 2 permutations ......Page 272
1. Introduction ......Page 277
2. Group Action on Weierstrass Points ......Page 278
3. Equations and Group Generators ......Page 281
