Families of twists of tuples of hyperelliptic curves

Official URL: https://dx.doi.org/10.1017/S0017089525100864

Let f ∈ Q[x] be a square-free polynomial of degree at least 3, mi, i = 1, 2, 3, odd positive integers, and ai, i = 1, 2, 3, non-zero rational numbers. We show the existence of a rational function D ∈ Q(v1, v2, v3, v4)such that the Jacobian of the quadratic twist of y2 = f(x) and the Jacobian of the mi-twist, respectively, 2mi-twist, of y2 = xmi + a2i , i = 1, 2, 3, by D are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.