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A modular approach to cubic Thue-Mahler equations

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Title
A modular approach to cubic Thue-Mahler equations
Author(s)
Dohyeong Kim
Publication Date
2017-05
Journal
MATHEMATICS OF COMPUTATION, v.86, no.305, pp.1435 - 1471
Publisher
AMER MATHEMATICAL SOC
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 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. © 2016 American Mathematical Society
URI
https://pr.ibs.re.kr/handle/8788114/3424
ISSN
0025-5718
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
Files in This Item:
S0025-5718-2016-03139-7.pdfDownload

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