Mahler’s work on Diophantine equations and subsequent developments

  • Jan-Hendrik Evertse

    Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands
  • Kálmán Győry

    Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
  • Cameron L. Stewart

    Dept. 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.