منحنی‌های جبری و کاربردهای آن‌ها

Families of elliptic curves with rational torsion points of even order / Boris M. Bekker and Yuri G. Zarhin -- The weighted moduli space of binary sextics / Lubjana Beshaj and Scott Guest -- A family of nonnormal double planes associated to hyperelliptic curvess / Tim J. Ford -- On the discriminant of a certain quadrinomials / Shuichi Otake and Tony Shaska -- Semistable types of hyperelliptic curves / Tim Dokchitser, Vladimir Dokchitser, Céline Maistret, and Adam Morgan -- Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function / Andrew R. Stout -- Computing heights on weighted projective spaces / Jorgo Mandili and Tony Shaska -- On hyperelliptic curves of genus 3 / Lubjana Beshaj and Monika Polak -- On automorphisms of algebraic curves / Allen Broughton, Tony Shaska, and Aaron Wootton -- On the algebraic classifcation of subgroups of hyperebolic planar crystallographic groups / Ismael Cortazaar and Antonio F. Costa -- On regular dessins d'enfants with 4g automorphisms and a curve of Wiman / Emilio Bujalance, Marston D.E. Conder, Antonio F. Costa, and Milagros Izquierdo -- An explicit descent of real algebraic varieties / Rubén A. Hidalgo -- Curves in isomonodromy and isospectral deformations: Painlevé as a case study / Emma Previato -- Quasi-quadratic residue codes and hyperelliptic curves / Nigel Boston and Jing Hao -- Curves, Jacobians, and cryptography / Gerhard Frey and Tony Shaska. Cover 1 Title page 4 Contents 6 Preface 8 Families of elliptic curves with rational torsion points of even order 12 1. Introduction 12 2. Review of [1] 14 3. Division by 2 16 4. Another criterion of divisibility by 2 21 5. Elliptic curves with points of order 4 26 6. Elliptic curves with points of order 8 28 7. Elliptic curves with point of order 6 32 8. Elliptic curves with point of order 12 35 9. Elliptic curves with rational points of order 10 37 10. Elliptic curves in characteristic 2 38 References 43 The weighted moduli space of binary sextics 44 1. Introduction 44 2. Preliminaries 46 3. Rational points in the weighted moduli space 50 Acknowledgment 54 References 54 A family of nonnormal double planes associated to hyperelliptic curves 56 1. Introduction 56 2. Background material 58 3. The main theorem 60 4. Examples and applications 62 References 64 On the discriminant of certain quadrinomials 66 1. Introduction 66 2. Preliminaries 67 3. Computation of the matrix A_{\scriptsize\boldmathc}(t)_{n-2} 69 4. Proof of theorems 78 5. Concluding remarks 80 References 81 Semistable types of hyperelliptic curves 84 1. Introduction 84 2. Background on metric graphs 89 3. Hyperelliptic graphs, BY trees and cluster pictures 91 4. One-to-one correspondence (open case) 101 5. One-to-one correspondence (closed case) 115 6. The homology lattice Λ 125 7. Tamagawa groups of hyperelliptic graphs 131 8. Classification of semistable types and naming convention 137 9. Classification for small genera 141 Acknowledgements 145 References 145 Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function 148 1. Introduction 148 2. Representability results concerning Hom 149 3. Projective systems of Hilbert spaces 150 4. Finite free algebras 151 5. Auto-arc spaces of formal deformations 153 6. A generalization of the motivic Igusa zeta function 154 7. Motivic rationality for plane curve singularities 156 References 157 Computing heights on weighted projective spaces 160 1. Introduction 160 2. Weight projective spaces 162 3. Heights on the weighted projective spaces 166 4. Computing the weighted height 168 References 169 On hyperelliptic curves of genus 3 172 1. Introduction 172 2. Preliminaries 173 3. Weighted moduli space of binary octavics 176 4. A database of genus 3 hyperelliptic curves 179 5. Concluding remarks 180 Acknowledgment 180 References 183 On automorphisms of algebraic curves 186 1. Introduction 186 2. Algebraic curves and their function fields 188 3. Weierstrass Gap theorem and Weierstrass points 194 4. Automorphisms of curves 197 5. Superelliptic curves 200 6. Automorphism groups of compact Riemann surfaces 206 References 220 On the algebraic classification of subgroups of hyperbolic planar crystallographic groups 224 1. Introduction 224 2. Canonical presentation 227 3. Hoare’s algorithm to compute de signature of a subgroup 227 4. Examples where direction of period-cycles is essential to know if two subgroups of a hyperbolic crystallographic group are isomorphic 229 5. Computing the signature of subgroups using a package of GAP 232 Acknowledgments 233 References 233 On regular dessins d’enfants with 4g automorphisms and a curve of Wiman 236 1. Introduction 236 2. Background 237 3. Regular dessins d’enfants with 4g automorphisms 239 References 243 An explicit descent of real algebraic varieties 246 1. Introduction 246 2. Main results 247 3. Proof of Theorems 1 and 2 249 4. An example 253 5. MAGMA implementation 255 Acknowledgments 256 References 256 Curves in isomonodromy and isospectral deformations: Painlevé VI as a case study 258 1. Introduction 258 2. Connecting integrable PDEs and Painlevé-type. I: Isomonodromy aspects 261 3. Connecting integrable PDEs and Painlevé-type. II: Symmetry reduction 266 4. Comments and further aspects 272 Acknowledgment 273 References 273 Quasi-quadratic residue codes and hyperelliptic curves 278 1. Quasi-quadratic residue codes 278 2. Hyperelliptic curves 283 References 289 Curves, Jacobians, and cryptography 290 Preface 290 Part 1. Abelian varieties 293 1. Definitions and basic properties 293 2. Endomorphisms and isogenies 297 3. Projective curves and Jacobian varieties 302 4. Applications of the Riemann-Roch theorem 307 5. Modular curves 325 Part 2. Cryptography 328 6. Diffie-Hellman key exchange 328 7. Index calculus in Picard groups 332 8. Isogenies of Jacobians via correspondences and applications to discrete logarithms 335 9. Genus 3 curves and cryptography 338 10. Genus 2 curves and cryptography 342 11. Elliptic curve cryptography 348 References 352 Back Cover 358 Families of elliptic curves with rational torsion points of even order / Boris M. Bekker and Yuri G. Zarhin -- The weighted moduli space of binary sextics / Lubjana Beshaj and Scott Guest -- A family of nonnormal double planes associated to hyperelliptic curves / Timothy J. Ford -- On the discriminant of certain quadrinomials / Shuichi Otake and Tony Shaska -- Semistable types of hyperelliptic curves / Tim Dokchitser, Vladimir Dokchitser, Céline Maistret and Adam Morgan -- Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function / Andrew R. Stout -- Computing heights on weighted projective spaces / Jorgo Mandili and Tony Shaska -- On hyperelliptic curves of genus 3 / L. Beshaj and M. Polak -- On automorphisms of algebraic curves / A. Broughton, T. Shaska and A. Wootton -- On the algebraic classification of subgroups of hyperbolic planar crystallographic groups / Ismael Cortázar and Antonio F. Costa -- On regular dessins d'enfants with 4g automorphisms and a curve of Wiman / Emilio Bujalance, Marston D. E. Conder, Antonio F. Costa and Milagros Izquierdo -- An explicit descent of real algebraic varieties / Rubén A. Hidalgo -- Curves in isomonodromy and isospectral deformations: Painlevé VI as a case study / E. Previato -- Quasi-quadratic residue codes and hyperelliptic curves / Nigel Boston and Jing Hao -- Curves, Jacobians, and cryptography / Gerhard Frey and Tony Shaska "This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography. This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics."--Publisher's description This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, h. Read more...