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

TL;DR

This paper investigates the arithmetic properties of a family of elliptic curves with specific discriminants, exploring their rational points, Selmer groups, and genus-one curves that violate the Hasse principle, under the assumption of finiteness of the Tate-Shafarevich group.

Contribution

It establishes new bounds on the ranks of these elliptic curves, relates their Selmer groups to their ranks, and constructs explicit genus-one curves violating the Hasse principle.

Findings

Proves bounds for the rank of the elliptic curves.

Establishes a parity relation between the rank and 3-Selmer group.

Constructs explicit genus-one curves 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.