By J. E. Cremona

This book is in three sections. First, we describe 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 $X_0(N)$, which are attached to certain cusp forms for the congruence subgroup $\Gamma_0(N)$. 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, we report on the results obtained when the modular symbols algorithm was carried out for all $N\le1000$. 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. [In the first edition, these numbers were given as 5089 and 2447 respectively, as the curves of conductor 702 were inadvertently omitted.] Specifically, we give for each curve 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 $s=1$; we also give the reduction data (Kodaira symbols, and local constants) for all primes of bad reduction, and information about isogenies..........

Contents of Second Edition

- Introduction
- Modular symbol algorithms
- Modular Symbols and Homology
- The upper half-plane, the modular group and cusp forms
- The duality between cusp forms and homology
- Real structure
- Modular symbol formalism
- Rational structure and the Manin-Drinfeld Theorem
- Triangulations and homology
- M-symbols and $\Gamma_0(N)$
- Conversion between modular symbols and M-symbols
- Action of Hecke and other operators
- Working in $H^+(N)$
- Modular forms and modular elliptic curves
- Splitting off one-dimensional eigenspaces
- $L(f,s)$ and the evaluation of $L(f,1)/\period(f)$
- Computing Fourier coefficients
- Computing periods I
- Computing periods II: Indirect method
- Computing periods III: Evaluation of the sums
- Computing $L^{(r)}(f,1)$
- Obtaining equations for the curves
- Computing the degree of a modular parametrization
- Modular Parametrizations
- Coset representatives and Fundamental Domains
- Implementation for $\Gamma_0(N)$ Appendix to Chapter II. Examples
- Example 1. N=11
- Example 2. N=33
- Example 3. N=37
- Example 4. N=49
- Elliptic curve algorithms
- Terminology and notation
- The Kraus--Laska--Connell algorithm and Tate's algorithm
- The Mordell--Weil group I: finding torsion points
- Heights and the height pairing
- The Mordell--Weil group II: generators
- The Mordell--Weil group III: the rank
- The period lattice
- Finding isogenous curves
- Twists and complex multiplication
- The tables
- Elliptic curves
- Mordell--Weil generators
- Hecke eigenvalues
- Birch--Swinnerton-Dyer data
- Parametrization degrees
- Bibliography