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In this paper it is shown that, given n ∈ Z ≥3 and a, b, c ∈ Q × , there exists a polynomial d(t) ∈ Q[t] such that the curves over Q(t) given by y 2 = x n + ad(t) resp. y 2 = x n + bd(t) and y 2 = x n + cd(t) all have a Jacobian with positive Mordell-Weil rank over Q(t). Extensions of this result to sets of four curves are discussed, as well as the problem of demanding in addition that d(t) is a square.

Tuples of hyperelliptic curves y2=xn+a

Tuples of hyperelliptic curves y²=xⁿ+a