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= ?,where the parameter t, the root of unity? 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 ifRet=?1/2 (there are a few more solutions in this case which are explicitly listed). In the caseRet=?1/2, an algebraic method is used, in the general case, Baker?s method yields the result.

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