Ranks of Jacobians of Curves Related to Binary Forms

Rubin and Silverberg reformulated the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded into the question of whether a certain infinite series converges. Da¸browski and Je¸drzejak consider an analogue of this theorem for the family of Jacobian varieties of twisted Fermat curves x^p + y^p = m (p fixed odd prime), where m runs through pth power-free integers. In this paper, we consider the family of Jacobian varieties of curves f (x, y) = mzⁿ, where f (x, y) ∈ Z[x, y] is an irreducible binary form of degree n ≥ 3 and m is an nth power-free integer.