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 λ-isogenous curves $E_{D^{\prime }}:y^2=x^3-27\cdot 16D$, where D and $D^{\prime }=-3D$ are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field $K_D=Q\left(\sqrt{D}\right)$ always has non-trivial 3-class group. We prove a series of results related to the set of rational points $E_{D^{\prime }}\left(Q\right)\setminus \lambda \left(E_D\right(Q\left)\right)$, and the $SL(2,Z)$-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.