Algorithms for Modular Elliptic Curves

J. E. Cremona

Contents of Second Edition

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

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