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Congruent numbers and congruences between half-integral weight modular forms

In this paper we investigate 2-parts of class numbers of quadratic imaginary field Q(sqrt(-d)) and 2-parts of the algebraic parts of the central L-values associated to the elliptic curves E_d : y^2 = x^3-d^2x 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 E_d, we prove results about the structure of the 2-part of the Tate-Shafarevich group X(E_d). Bruin and Hemenway unconditionally proved some of these results, therefore we verify that for curves E_d Birch and Swinnerton-Dyer conjecture gives correct predictions about the size of 2-part of its Tate-Shafarevich group.

congruences between modular forms ; congruent numbers ; class numbers ; Birch and Swinnerton-Dyer conjecture

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