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

Abstract

We study an infinite family of $j$-invariant zero elliptic curves $E_{D}:y^{2}=x^{3}+16D$ and their $\lambda$-isogenous curves $E_{D'}:y^{2}=x^{3}-27\cdot16D$, where $D$ and $D' = -3D$ are fundamental discriminants of a specific form, and $\lambda$ is an isogeny of degree $3$. A result of Honda guarantees that for our discriminants $D$, the quadratic number field $K_{D} = \mathbb{Q}(\sqrt{D})$ always has non-trivial 3-class group. We prove 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 $D$. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of $E_{D}$ and the rank of its $3$-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-$1$ curves that correspond to irreducible integral binary cubic forms of discriminant $D=48035713$, and we show that every curve in these classes violates the Hasse Principle.