A relation between the value of the L-series of the curve: \(y^ 2=x^ 3-k^ 3\) in \(s=1\) and its Selmer group
Let \(E_ k\) be the elliptic curve defined over \({\mathbb{Q}}\) by the equation \(y^ 2=x^ 3-k^ 3,\) where \(k\equiv 1 (mod 4)\), squarefree and prime to 6. The author verifies a weak form of the Birch and Swinnerton-Dyer conjecture on the value of the L-series of the curve \(E_ k\) at \(s=1\) [see \textit{J. Coates} and \textit{A. Wiles}, Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)]. In fact he proves that the k-th Fourier coefficient of a well-known related modular form of weight 3/2 is divisible by 3 if and only if the Tate-Shafarevich group of \(E_ k\) has elements of order 3. By means of Galois cohomology it is shown that the latter is the case if and only if the order of the class group of \({\mathbb{Q}}(\sqrt{-k})\) is divisible by 3.
scientific article; zbMATH DE number 3918549