Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

摘要

We consider the family of relative Thue equations x~3 - (t - 1)x~2y - (t + 2)xy~2 - y~3 = μ, where the parameter t, the root of unity μ 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 Re t = - 1/2 (there are a few more solutions in this case which are explicitly listed). In the case Re t = -1/2, an algebraic method is used, in the general case, Baker's method yields the result.