Variations on twists of tuples of hyperelliptic curves and related results

Original · EN

Let f∈[x] be a square-free polynomial of degree ≥ 3 and m≥ 3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D₁∈(u,v,w) such that the Jacobians of the curves equation* C₁:D₁y²=f(x), C₂:y²=D₁xᵐ+b, C₃:y²=D₁xᵐ+c, equation* have all positive ranks over (u,v,w). Similarly, we prove that there exists a function D₂∈(u,v,w) such that the Jacobians of the curves equation* C₁:D₂y²=h(x), C₂:y²=D₂xᵐ+b, C₃:y²=xᵐ+cD₂, equation* have all positive ranks over (u,v,w). Moreover, if f(x)=xᵐ+a for some a∈{0}, we prove the existence of a function D₃∈(u,v,w) such that the Jacobians of the curves equation* C₁:y²=D₃xᵐ+a, C₂:y²=D₃xᵐ+b, C₃:y²=xᵐ+cD₃, equation* have all positive ranks over (u,v,w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves yᵖ=xᵐ(x+a) and yᵖ=xᵐ(a-x)ᵏ for a prime p and 0<m<p-2 and k<p and apply our result in order to prove the existence of a function D∈(u,v,w,t) such that the Jacobians of the curves equation* C₁:Dyᵖ=xᵐ(x+a), Dyᵖ=xᵐ(x+b) equation* have both positive rank over (u,v,w,t).