Computational aspects of algebraic curves : [proceedings] / editor, Tanush Shaska.

Format:

Book

Conference/Event

Author/Creator:

Conference on Computational Aspects of Algebraic Curves, Corporate Author.

Contributor:

Shaska, Tony, 1967-

Conference Name:

Conference on Computational Aspects of Algebraic Curves (2005 : University of Idaho)

Conference on Computational Aspects of Algebraic Curves

Series:

Lecture Notes Series on Computing

Lecture notes series on computing ; v. 13

Language:

English

Subjects (All):

Curves, Algebraic--Data processing--Congresses.

Curves, Algebraic.

Geometry, Algebraic--Data processing--Congresses.

Geometry, Algebraic.

Physical Description:

1 online resource (286 p.)

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, c2005.

Language Note:

English

Summary:

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 cove

Contents:

CONTENTS; 1. Preface; 2. Foreword by the Editor; 3. A new proof for the non-degeneracy of the Frey-Ruck pairing and a connection to isogenies over the base field; 1. Introduction; 2. Overview of mathematical methods; 3. Description of the first pairing; 4. Proof of non-degeneracy of the first pairing; 5. Motivation for the pairing using isogenies over the base field; 6. Applications to elliptic curves; 4. Elliptic curve torsion points and division polynomials; 1. Introduction; 2. Division Polynomials; 3. E(Q)tors and Hensel's lemma; 4. Deciding whether E(Qp)[p] is non-trivial

5. Computing E(Q)[2]6. The p-adic algorithm; 7. Discriminant of the division polynomial; 8. The l-adic algorithm; 9. Time complexity analysis of the algorithms; 10. Conclusions; 5. Detecting complex multiplication; 1. Introduction; 2. Background; 3. Algorithms for elliptic curves; 4. Algorithmic considerations; 6. Simple numerical uniformatization of elliptic curves; 1. Myrberg's algorithm for elliptic curves; 2. Generalizations of Myrberg's algorithm; 7. On the moduli space of Klein four covers of the projective line; 1. Introduction; 2. Hurwitz Spaces; 3. The geometry of Hg2

4. The geometry of Hg38. Field of moduli and field of definition for curves of genus 2; 1. Preliminaries on genus 2 curves; 2. The case Aut(C) - V4; 3. The case Aut(C)-D8 D12; 4. The isolated cases; 9. Explicit computation of Hurwitz spectra; 1. Introduction; 2. The combinatorial technique; 3. Computing the traces; 4. Structure of the trace sequence; 10. Non-normal Belyi p-gonal surfaces; 1. Introduction; 2. Fuchsian Groups and Uniformization; 3. Finding Signatures; 4. Finding Non-Normal Belyi Surfaces; 11. Hyperelliptic curves of genus 3 with prescribed automorphism group; 1. Introduction

2. Dihedral invariants of hyperelliptic curves3. Hyperelliptic curves of genus three; 4. Field of moduli of genus 3 hyperelliptic curves; 12. Curves over finite fields with many points: an introduction; 1. Introduction; 2. Goppa codes; 3. Upper bounds; 4. Lower bounds; 5. Determination of Nq(g); 13. Hyperelliptic curves of genus 3 and 4 in characteristic 2; 1. Introduction; 2. Preliminaries; 3. Hyperelliptic curves of genus 3; 4. Hyperelliptic curves of genus 4; 5. APPENDIX; 14. Modular representations on some Riemann-Roch spaces of modular curves X(N); 1. Introduction; 2. Modular curves

3. Representation theory of PSL(2 N)4. Induced characters; 5. Ramification module; 6. Equivariant degree and Riemann-Roch space; 7. Examples; 8. Application to codes; 9. A G codes associated to X(7); 10. Appendix A: Tables for Theorem 9; 11. Appendix B: Ramification modules and GAP code; 15. Genus two curves covering elliptic curves: a computational approach; 1. Introduction; 2. Preliminaries; 3. Curves of genus 2 with split Jacobians; 4. The locus of genus two curves with (n n) split Jacobians; 5. Genus 2 curves with degree 2 elliptic subcovers

6. Genus 2 curves with degree 3 elliptic subcovers

Notes:

"University of Idaho, USA, 26-28 May 2005."

Includes bibliographical references.

ISBN:

9786611897253

9781281897251

1281897256

9789812701640

9812701648