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.

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Mathematics

**Publication date**: 31 Dec 1997

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**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.

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