**Contemporary Mathematics**

Volume: 724;
2019;
344 pp;
Softcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-4247-7
Product Code: CONM/724**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-5153-0
Product Code: CONM/724.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# Algebraic Curves and Their Applications

Share this page *Edited by *
*Lubjana Beshaj; 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.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry, cryptography, and elliptic and hyperelliptic curves.

# Table of Contents

## Algebraic Curves and Their Applications

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Families of elliptic curves with rational torsion points of even order 112
- 1. Introduction 112
- 2. Review of [1] 314
- 3. Division by 2 516
- 4. Another criterion of divisibility by 2 1021
- 5. Elliptic curves with points of order 4 1526
- 6. Elliptic curves with points of order 8 1728
- 7. Elliptic curves with point of order 6 2132
- 8. Elliptic curves with point of order 12 2435
- 9. Elliptic curves with rational points of order 10 2637
- 10. Elliptic curves in characteristic 2 2738
- References 3243

- The weighted moduli space of binary sextics 3344
- A family of nonnormal double planes associated to hyperelliptic curves 4556
- On the discriminant of certain quadrinomials 5566
- Semistable types of hyperelliptic curves 7384
- 1. Introduction 7384
- 2. Background on metric graphs 7889
- 3. Hyperelliptic graphs, BY trees and cluster pictures 8091
- 4. One-to-one correspondence (open case) 90101
- 5. One-to-one correspondence (closed case) 104115
- 6. The homology lattice Λ 114125
- 7. Tamagawa groups of hyperelliptic graphs 120131
- 8. Classification of semistable types and naming convention 126137
- 9. Classification for small genera 130141
- Acknowledgements 134145
- References 134145

- Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function 137148
- 1. Introduction 137148
- 2. Representability results concerning Hom 138149
- 3. Projective systems of Hilbert spaces 139150
- 4. Finite free algebras 140151
- 5. Auto-arc spaces of formal deformations 142153
- 6. A generalization of the motivic Igusa zeta function 143154
- 7. Motivic rationality for plane curve singularities 145156
- References 146157

- Computing heights on weighted projective spaces 149160
- On hyperelliptic curves of genus 3 161172
- On automorphisms of algebraic curves 175186
- On the algebraic classification of subgroups of hyperbolic planar crystallographic groups 213224
- 1. Introduction 213224
- 2. Canonical presentation 216227
- 3. Hoare’s algorithm to compute de signature of a subgroup 216227
- 4. Examples where direction of period-cycles is essential to know if two subgroups of a hyperbolic crystallographic group are isomorphic 218229
- 5. Computing the signature of subgroups using a package of GAP 221232
- Acknowledgments 222233
- References 222233

- On regular dessins d’enfants with 4g automorphisms and a curve of Wiman 225236
- An explicit descent of real algebraic varieties 235246
- Curves in isomonodromy and isospectral deformations: Painlevé VI as a case study 247258
- Quasi-quadratic residue codes and hyperelliptic curves 267278
- Curves, Jacobians, and cryptography 279290
- Preface 279290
- Part 1. Abelian varieties 282293
- Part 2. Cryptography 317328
- 6. Diffie-Hellman key exchange 317328
- 7. Index calculus in Picard groups 321332
- 8. Isogenies of Jacobians via correspondences and applications to discrete logarithms 324335
- 9. Genus 3 curves and cryptography 327338
- 10. Genus 2 curves and cryptography 331342
- 11. Elliptic curve cryptography 337348
- References 341352

- Back Cover Back Cover1358