Contents

• The BSD Rank Conjecture
  • Statement of the BSD Rank Conjecture
  • The BSD Rank Conjecture Implies that $E(\mathbb{Q})$ is Computable
  • The Complex $L$-series $L(E,s)$
  • Computing $L(E,s)$
    • Approximating the Rank
  • The $p$-adic $\mathcal{L}$-series
    • Hensel's lemma and the Teichmuller lift
    • Modular Symbol and Measures
    • The $p$-Adic $L$-function
  • Computing $\mathcal{L}_p(E,T)$
  
  
• The Birch and Swinnerton-Dyer Formula
  • Galois Cohomology
    • Group Cohomology
    • The inf-res Sequence
    • Galois Cohomology
  • The Shafarevich-Tate Group
    • The Elliptic Curve Kummer Sequence
    • The Global-to-Local Restriction Maps
    • The Selmer Group
    • The Shafarevich-Tate Group and the Mordell-Weil Theorem
    • Some Conjectures and Theorems about the Shafarevich-Tate Group
  • The Birch and Swinnerton-Dyer Formula
  • Examples: The Birch and Swinnerton-Dyer Formula
    • Example: A Curve of Rank 0
    • Example: A Rank 0 curve with nontrivial Sha
    • Example: A Curve of Rank 1
    • Example: A curve of rank $2$
    • Example: A Rank 3 curve
    • Example: A Rank 4 curve
  • The $p$-adic BSD Conjectural Formula
    • Example: A Curve of Rank $2$
    • The $p$-adic Regulator
  
  
• Heegner Points and Kolyvagin's Euler System
  • CM Elliptic Curves
    • The Set of CM Elliptic Curves with Given CM
    • Class Field Theory
    • The Field of Definition of CM Elliptic Curves
  • Heegner Points
  • Computing Heegner Points
  • Kolyvagin's Euler System
    • Kolyvagin's Cohomology Classes
    • Kolyvagin's Conjectures
  • The Gross-Zagier Theorem
  
  
• Computational Verification of the Conjecture
  • Theorem
  • Examples
  
  
• Bibliography
• Index