A modular approach to cubic Thue-Mahler equations

Kim, Dohyeong

doi:10.1090/mcom/3139

A modular approach to cubic Thue-Mahler equations
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Math. Comp. 86 (2017), 1435-1471 Request permission

Abstract:

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 $\mathbb {Q}$ 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.

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