Exact classification of elliptic curves $y^2=x^3-pqx$ with rank $0$ and trivial $\Sha[2]$

Abstract

For the elliptic curves $E_{p,q}:y^2=x^3-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $\varphi$ and its dual $\hat{\varphi }$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.

initial submitted version. Comments very much welcomed!