On certain twisted families of elliptic curves of rank 8 (Q1382630)

The author constructs infinite families of elliptic curves defined over \(\mathbb{Q}\) of rank \(\geq 8\). The structure of the construction is the following. Let \(K\) be a number field, and fix an elliptic curve \(E\) defined over \(K\) with the following equation: \[ y^2=x^3+ax^2+ax+1 \] (\(a\) belongs to \(K\)). Let \(C\) be the hyperelliptic curve over \(K\) given by the equation \[ y^2=x^6+ax^4+ax^2+1; \] it has an involution \(\gamma(x,y)=(x,-y)\). Let \(V\) be the quotient of \(C^4\) by all the involutions \(\gamma_i\) for \(i=1,\ldots,4\). The author shows that the twist \(E'\) of \(E\) associated to the quadratic extension \(K(C^4)/K(V)\) is such that \(E'(K(V))\) has \(8\) points of infinite order which are also \(\mathbb{Z}\)-independent. These points are computed explicitly. The author shows also that there are infinitely many \(K\)-rational points on \(V\). The author finishes his construction by putting \(K=\mathbb{Q}\) and using an argument of specialisation. The use of the twist theory goes back to \textit{F. Hazama} [|J. Number Theory 50, No. 2, 278-285 (1995; Zbl 0838.14024)].