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A modular approach to cubic Thue-Mahler equations


Author: Dohyeong Kim
Journal: Math. Comp. 86 (2017), 1435-1471
MSC (2010): Primary 11D59, 11F11, 11Y50
DOI: https://doi.org/10.1090/mcom/3139
Published electronically: September 15, 2016
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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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Additional Information

Dohyeong Kim
Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), 77 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 790-784, Republic of Korea – and – Pohang University of Science and Technology (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang-si, Gyeongsangbuk-do, 790-784, Republic of Korea
Address at time of publication: Department of Mathematics, University of Michigan, 2074 East Hall, Ann Arbor, Michigan 48109-1043
Email: dohyeong@umich.edu

DOI: https://doi.org/10.1090/mcom/3139
Received by editor(s): June 9, 2015
Received by editor(s) in revised form: November 27, 2015
Published electronically: September 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society