Mahler’s work on Diophantine equations and subsequent developments

Abstract

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 $p$-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 $S$-integers, for any arbitrary finite set of primes $S$. 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 $S$-integral points. In this chapter, we discuss Mahler's work on Diophantine approximation and its applications to Diophantine equations, in particular Thue-Mahler equations, $S$-unit equations and $S$-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 $P$-adic logarithmic forms, which may be viewed as an outgrowth of Mahler's work on the $P$-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.