Congruent numbers and congruences between half-integral weight modular forms

In this paper we investigate 2-parts of class numbers of quadratic imaginary field and 2-parts of the algebraic parts of the central L-values associated to the elliptic curves Ed:y2=x3−d2x by studying congruences modulo small powers of two between certain half-integral weight modular forms. Assuming the full Birch and Swinnerton–Dyer conjecture for elliptic curves Ed, we prove results about the structure of the 2-part of the Tate–Shafarevich group Ш(Ed). Bruin and Hemenway (2011) [3] unconditionally proved some of these results, therefore we verify that for curves Ed Birch and Swinnerton–Dyer conjecture gives correct predictions about the size of 2-part of its Tate–Shafarevich group.