Project: to study the TateShafarevich groups of elliptic curves
E_{d} : y^{2} = x^{3}  d^{2} x for squarefree integers d.
Method:

Use Tunnell's Theorem [1] and the Birch and SwinnertonDyer conjecture (p. 330 of [1]) to compute the conjectured order of Sha(E_d) when L(E_d,1) is nonzero.

Use pp. 25/26 of [2] and a 2descent (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 2part of Sha(E_d).
References:

Tunnell, J.: A classical diophantine problem and modular forms of weight 3/2. Inventiones math. 72 (1983) 323334.

Rubin, K.: The "main conjectures" of Iwasawa theory for imaginary quadratic fields. Inventiones math. 103 (1991) 2568. For an easier (but less general) exposition see http://math.stanford.edu/~rubin/cime.dvi

Silverman, J.: The Arithmetic of elliptic curves. Graduate texts in Math. 106, SpringerVerlag (1986)
