Thomas’ Family of Thue Equations Over Imaginary Quadratic Fields

We consider the family of relative Thue equations x3 − (t − 1)x2y− (t + 2)xy2 − y3 = μ,where the parameter t, the root of unityμ and the solutions x and y are integers in the same imaginary quadratic number field. We prove that there are only trivial solutions (with x, y ≤ 1), if t is large enough or if the discriminant of the quadratic number field is large enough or if t = − 1 / 2 (there are a few more solutions in this case which are explicitly listed). In the case t = − 1 / 2, an algebraic method is used, in the general case, Baker’s method yields the result.