Efficient computation of rank of elliptic curves using hyperelliptic curves associated with the simplest cubic fields (Q5928553)

The simplest cubic field \(K\) is defined as the splitting field of the irreducible polynomial \(f(X)=X^3+mX^2-(m+3)X+1\) over \({\mathbb{Q}}\) introduced by D. Shanks. The author considers the elliptic curves \(E\) \(Y^2=f(X)\). Let \(C\) be the class field of \(K\), \(C_2=\{x\in C\mid x^2=1\}\), \(\text{ Ш}_2\) the \(2\)-torsion of the Tate-Shafarevich group of \(E\) and \(E^0\) the right hand side connected component of \(E\) over \(\mathbb{Q}\). Suppose that the discriminant \(D\) of \(f(X)\) is square-free. Then there exists an exact sequence due to L. Washington NEWLINE\[NEWLINE1\to E^0(\mathbb{Q})/2E(\mathbb{Q}) @>{\mu}>> C_2\to\text{ Ш}_2\to 1,NEWLINE\]NEWLINE where \(x-\rho=I^2\), \(\rho<0\) is a root of \(f(X)\) and \(\mu((x,y))=I\). As a consequence, \({\text{rank}}(E(\mathbb{Q}))\leq 1+{\text{rk}}_2(C_2)\).NEWLINENEWLINENEWLINEThe author considers the following genus \(2\) curve \(\widehat{E}:y^2=z^6+(m+3)z^4+mz^2-1\) and proves that if \((z,y)\in\widehat{E}(\mathbb{Q})\), then \(P=(z^2=1,y)\in E(\mathbb{Q})\), \(P'=(-\frac 1{z^2},\frac y{z^3})\in E(\mathbb{Q})\). Let \((z^2+1-\rho)=I^2\) for some ideal \(I\) of \(K\), then \((-\frac 1{z^2}-\rho)=(I')^2\), where \(I'\) is the conjugate of \(I\). In particular, if \(P\notin E(\mathbb{Q})\) then \({\text{rank}}_2(C_2)\geq 2\) and \({\text{rank}}(E(\mathbb{Q}))\geq 3\).