Algorithms for Modular Elliptic Curves, Second Edition

Describes in detail an algorithm based on modular symbols for computing modular elliptic curves. Also describes various algorithms for studying the arithmetic of elliptic curves.

**Tag(s):**
Mathematics

**Publication date**: 31 Dec 1997

**ISBN-10**:
n/a

**ISBN-13**:
n/a

**Paperback**:
n/a

**Views**: 16,090

**Type**: N/A

**Publisher**:
n/a

**License**:
n/a

**Post time**: 26 Dec 2006 01:58:13

Algorithms for Modular Elliptic Curves, Second Edition

Describes in detail an algorithm based on modular symbols for computing modular elliptic curves. Also describes various algorithms for studying the arithmetic of elliptic curves.

Book Excerpts:

This book is in three sections. First, it describes in detail an algorithm based on modular symbols for computing modular elliptic curves: that is, one-dimensional factors of the Jacobian of the modular curve Xo(N), which are attached to certain cusp forms for the congruence subgroup To(N).

In the second section, various algorithms for studying the arithmetic of elliptic curves (defined over the rationals) are described. These are for the most part not new, but they have not all appeared in book form, and it seemed appropriate to include them here.

Lastly, this book reports on the results obtained when the modular symbols algorithm was carried out for all N <= 1000. In a comprehensive set of tables we give details of the curves found, together with all isogenous curves (5113 curves in all, in 2463 isogeny classes). Specifically, each curve is given the rank and generators for the points of infinite order, the number of torsion points, the regulator, the traces of Frobenius for primes less than 100, and the leading coefficient of the L-series at s = 1; this book also shows the reduction data (Kodaira symbols, and local constants) for all primes of bad reduction, and information about isogenies.

Intended Audience:

This book does not intent to discuss the theory of modular forms in any detail, though there will be a summary of needed facts, and some references to suitable texts. The theoretical construction and properties of the modular elliptic curves will also be excluded, except for a brief summary. Likewise, this book will assume that the reader has some knowledge of the theory of elliptic curves, such as can be obtained from one of the growing number of excellent books on the subject. Instead this book will be concentrating on computational aspects, and thus complementing other, more theoretical, treatments.

This book is in three sections. First, it describes in detail an algorithm based on modular symbols for computing modular elliptic curves: that is, one-dimensional factors of the Jacobian of the modular curve Xo(N), which are attached to certain cusp forms for the congruence subgroup To(N).

In the second section, various algorithms for studying the arithmetic of elliptic curves (defined over the rationals) are described. These are for the most part not new, but they have not all appeared in book form, and it seemed appropriate to include them here.

Lastly, this book reports on the results obtained when the modular symbols algorithm was carried out for all N <= 1000. In a comprehensive set of tables we give details of the curves found, together with all isogenous curves (5113 curves in all, in 2463 isogeny classes). Specifically, each curve is given the rank and generators for the points of infinite order, the number of torsion points, the regulator, the traces of Frobenius for primes less than 100, and the leading coefficient of the L-series at s = 1; this book also shows the reduction data (Kodaira symbols, and local constants) for all primes of bad reduction, and information about isogenies.

Intended Audience:

This book does not intent to discuss the theory of modular forms in any detail, though there will be a summary of needed facts, and some references to suitable texts. The theoretical construction and properties of the modular elliptic curves will also be excluded, except for a brief summary. Likewise, this book will assume that the reader has some knowledge of the theory of elliptic curves, such as can be obtained from one of the growing number of excellent books on the subject. Instead this book will be concentrating on computational aspects, and thus complementing other, more theoretical, treatments.

Tweet

About The Author(s)

No information is available for this author.

Book Categories

Computer Science
Introduction to Computer Science
Introduction to Computer Programming
Algorithms and Data Structures
Artificial Intelligence
Computer Vision
Machine Learning
Neural Networks
Game Development and Multimedia
Data Communication and Networks
Coding Theory
Computer Security
Information Security
Cryptography
Information Theory
Computer Organization and Architecture
Operating Systems
Image Processing
Parallel Computing
Concurrent Programming
Relational Database
Document-oriented Database
Data Mining
Big Data
Data Science
Digital Libraries
Compiler Design and Construction
Functional Programming
Logic Programming
Object Oriented Programming
Formal Methods
Software Engineering
Agile Software Development
Information Systems
Geographic Information System (GIS)

Mathematics
Mathematics
Algebra
Abstract Algebra
Linear Algebra
Number Theory
Numerical Methods
Precalculus
Calculus
Differential Equations
Category Theory
Proofs
Discrete Mathematics
Theory of Computation
Graph Theory
Real Analysis
Complex Analysis
Probability
Statistics
Game Theory
Queueing Theory
Operations Research
Computer Aided Mathematics

Supporting Fields
Web Design and Development
Mobile App Design and Development
System Administration
Cloud Computing
Electric Circuits
Embedded System
Signal Processing
Integration and Automation
Network Science
Project Management

Operating System
Programming/Scripting
Ada
Assembly
C / C++
Common Lisp
Forth
Java
JavaScript
Lua
Microsoft .NET
Rexx
Perl
PHP
Python
R
Rebol
Ruby
Scheme
Tcl/Tk

Miscellaneous
Sponsors