Algebraic curves and their applications / Lubjana Beshaj, Tony Shaska, editors.

Format:

Book

Author/Creator:

Beshaj, Lubjana, 1986- editor.

Contributor:

Shaska, Tony, 1967- editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 724

Contemporary mathematics, 724 0271-4132

Language:

English

Subjects (All):

Curves, Algebraic.

Physical Description:

1 online resource (358 pages).

Edition:

1st ed.

Place of Publication:

[Place of publication not identified] : American Mathematical Society, [2019]

Language Note:

English

Summary:

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.

Contents:

Cover

Title page

Contents

Preface

Families of elliptic curves with rational torsion points of even order

1. Introduction

2. Review of [1]

3. Division by 2

4. Another criterion of divisibility by 2

5. Elliptic curves with points of order 4

6. Elliptic curves with points of order 8

7. Elliptic curves with point of order 6

8. Elliptic curves with point of order 12

9. Elliptic curves with rational points of order 10

10. Elliptic curves in characteristic 2

References

The weighted moduli space of binary sextics

2. Preliminaries

3. Rational points in the weighted moduli space

Acknowledgment

A family of nonnormal double planes associated to hyperelliptic curves

2. Background material

3. The main theorem

4. Examples and applications

On the discriminant of certain quadrinomials

3. Computation of the matrix _{\scriptsize\boldmath }( )_{ -2}

4. Proof of theorems

5. Concluding remarks

Semistable types of hyperelliptic curves

2. Background on metric graphs

3. Hyperelliptic graphs, BY trees and cluster pictures

4. One-to-one correspondence (open case)

5. One-to-one correspondence (closed case)

6. The homology lattice Λ

7. Tamagawa groups of hyperelliptic graphs

8. Classification of semistable types and naming convention

9. Classification for small genera

Acknowledgements

Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function

2. Representability results concerning Hom

3. Projective systems of Hilbert spaces

4. Finite free algebras

5. Auto-arc spaces of formal deformations

6. A generalization of the motivic Igusa zeta function.

7. Motivic rationality for plane curve singularities

Computing heights on weighted projective spaces

2. Weight projective spaces

3. Heights on the weighted projective spaces

4. Computing the weighted height

On hyperelliptic curves of genus 3

3. Weighted moduli space of binary octavics

4. A database of genus 3 hyperelliptic curves

On automorphisms of algebraic curves

2. Algebraic curves and their function fields

3. Weierstrass Gap theorem and Weierstrass points

4. Automorphisms of curves

5. Superelliptic curves

6. Automorphism groups of compact Riemann surfaces

On the algebraic classification of subgroups of hyperbolic planar crystallographic groups

2. Canonical presentation

3. Hoare's algorithm to compute de signature of a subgroup

4. Examples where direction of period-cycles is essential to know if two subgroups of a hyperbolic crystallographic group are isomorphic

5. Computing the signature of subgroups using a package of GAP

Acknowledgments

On regular dessins d'enfants with 4g automorphisms and a curve of Wiman

2. Background

3. Regular dessins d'enfants with 4 automorphisms

An explicit descent of real algebraic varieties

2. Main results

3. Proof of Theorems 1 and 2

4. An example

5. MAGMA implementation

Curves in isomonodromy and isospectral deformations: Painlevé VI as a case study

2. Connecting integrable PDEs and Painlevé-type. I: Isomonodromy aspects

3. Connecting integrable PDEs and Painlevé-type. II: Symmetry reduction.

4. Comments and further aspects

Quasi-quadratic residue codes and hyperelliptic curves

1. Quasi-quadratic residue codes

2. Hyperelliptic curves

Curves, Jacobians, and cryptography

Part 1. Abelian varieties

1. Definitions and basic properties

2. Endomorphisms and isogenies

3. Projective curves and Jacobian varieties

4. Applications of the Riemann-Roch theorem

5. Modular curves

Part 2. Cryptography

6. Diffie-Hellman key exchange

7. Index calculus in Picard groups

8. Isogenies of Jacobians via correspondences and applications to discrete logarithms

9. Genus 3 curves and cryptography

10. Genus 2 curves and cryptography

11. Elliptic curve cryptography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-5153-0