Rational points in geometric progressions on certain hyperelliptic curves

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y² = f(x), where f ∈ ℤ[x] is without multiple roots. We say that points P_i = (x_i; y_i) ∈ C(ℚ) for i = 1, 2, . ., k, are in geometric progression if the numbers x_i for i = 1, 2, . . ., k, are in geometric progression. Let n ≥ 3 be a given integer. In this paper we show that there exist polynomials a, b 2 ℤ[t] such that on the curve y² = a(t)xⁿ + b(t) (defined over the field ℚ(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y² = ax² + b. We also investigate for fixed b ∈ ℤ, when there can exist rationals y_i, i = 1, . . ., 4, with -y² _i - b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y² = ax + b which contain five points in geometric progression.