On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle (Q6633561)

Consider an infinite family of \(j\)-invariant zero elliptic curves \(E_D: y^2 = x^3+16D\) and their \(\lambda\)-isogeneous curves \(E_{D'}: y^2 = x^3 - 27\cdot 16D\), where \(D = D_{m,n}\) is a squarefree integer of the form \(4m^3 - 27n^2\), \(D' = -3D\), and \(\lambda\) is an isogeny of degree \(3\). A result of Honda guarantees that for such \(D\)'s, the quadratic field \(K_D = \mathbb Q(\sqrt{D})\) has non-trivial \(3\)-class group.\N\NThe author proves a series of results related to the set of rational points \(E_{D'}(\mathbb Q) \setminus \lambda(E_D(\mathbb Q))\), and the \(SL(2,\mathbb Z)\)-equivalence classes of irreducible integral binary cubic forms of discriminant \(D\). The main results are formulated in Proposition 1.2 (assuming finiteness of the Tate-Shafarevich group) and give:\N\N(i) a parity result between the rank of \(E_D\) and the rank of its \(3\)-Selmer group;\N\N(ii) lower and upper bounds for the rank of these elliptic curves.\N\NIn the last section, the author gives explicit classes of genus-\(1\) curves that correspond to irreducible integral binary cubic forms of discriminant \(D_{229,3}\), and shows that every curve in these classes violate the Hasse principle.