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A modular approach to cubic Thue-Mahler equations

Let h(x, y) be a non-degenerate binary cubic form with integral coefficients, and let S be an arbitrary finite set of prime numbers. By a classical theorem of Mahler, there are only finitely many pairs of relatively prime integers x, y such that h(x, y) is an S-unit. In the present paper, we reverse a well-known argument, which seems to go back to Shafarevich, and use the modularity of elliptic curves over Q to give upper bounds for the number of solutions of such a Thue-Mahler equation. In addition, our methods give an effective method for determining all solutions, and we use Cremona’s Elliptic Curve Database to give a wide range of numerical examples.