Number of solutions for cubic Thue equations with automorphisms

Abstract

Let \(F(X,Y)\in {\boldmath Z} [X,Y]\) be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation F(x,y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Pade approximation method developed in our former work. To obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained by geometric consideration.