Mahler’s work on Diophantine equations and subsequent developments
Jan-Hendrik EvertseMathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands
Kálmán GyőryInstitute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
Cameron L. StewartDept. of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
The main body of K. Mahler's work on Diophantine equations consists of his 1933 papers [Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01); 108, 37--55 (1933; Zbl 0006.15604); Acta Math. 62, 91--166 (1934; Zbl 0008.19801; JFM 60.0159.04)], in which he proved a generalization of the Thue-Siegel Theorem on the approximation of algebraic numbers by rationals, involving -adic absolute values, and applied this to get finiteness results for the number of solutions for what became later known as Thue-Mahler equations. He was also the first to give upper bounds for the number of solutions of such equations. In fact, Mahler's extension of the Thue-Siegel Theorem made it possible to extend various finiteness results for Diophantine equations over the integers to -integers, for any arbitrary finite set of primes . For instance Mahler himself [J. Reine Angew. Math. 170, 168--178 (1934; Zbl 0008.20002; JFM 60.0159.03)] extended Siegel's finiteness theorem on integral points on elliptic curves to -integral points.
In this chapter, we discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, -unit equations and -integral points on elliptic curves, and go into later developments concerning the number of solutions to Thue-Mahler equations and effective finiteness results for Thue-Mahler equations. For the latter we need estimates for -adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the -adic Gel'fond-Schneider theorem [Compos. Math. 2, 259--275 (1935; Zbl 0012.05302; JFM 61.0187.01)]. We also go briefly into decomposable form equations, these are certain higher dimensional generalizations of Thue-Mahler equations.