Arizona Winter School 1999
Rubin's Project Description

Project: to study the Tate-Shafarevich groups of elliptic curves

Ed : y2 = x3 - d2 x for squarefree integers d.


  1. Use Tunnell's Theorem [1] and the Birch and Swinnerton-Dyer conjecture (p. 330 of [1]) to compute the conjectured order of Sha(E_d) when L(E_d,1) is nonzero.
  2. Use pp. 25/26 of [2] and a 2-descent (Chapter 10, especially section 6 of [3]) to show that the conjectured order is the actual order in many cases, and to determine the rank of the 2-part of Sha(E_d).


  1. Tunnell, J.: A classical diophantine problem and modular forms of weight 3/2. Inventiones math. 72 (1983) 323-334.
  2. Rubin, K.: The "main conjectures" of Iwasawa theory for imaginary quadratic fields. Inventiones math. 103 (1991) 25-68. For an easier (but less general) exposition see
  3. Silverman, J.: The Arithmetic of elliptic curves. Graduate texts in Math. 106, Springer-Verlag (1986)