## Mahler's work on Diophantine equations and subsequent developments

##### Doc. Math. Extra Vol. Mahler Selecta, 149-171 (2019)
DOI: 10.25537/dm.2019.SB-149-171

### Summary

The main body of \textit{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. \par 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.

### Mathematics Subject Classification

11-03, 11D59, 11J61, 11J17

### References

• [M17]. K. Mahler, Zur Approximation algebraischer Zahlen, I: Über den größten Primteiler binärer Formen, Math. Ann. 107 (1933), 691-730.
• [M18]. K. Mahler, Zur Approximation algebraischer Zahlen, II: Über die Anzahl der Darstellungen ganzer Zahlen durch Binärformen, Math. Ann. 108 (1933), 37-55.
• [M19]. K. Mahler, Zur Approximation algebraischer Zahlen, III, Acta Math. 62 (1934), 91-166.
• [M21]. K. Mahler, Über rationalen Punkte auf Kurven vom Geschlecht Eins, J. Reine Angew. Math. (Crelle) 170 (1934), 168-178.
• [M28]. K. Mahler, On the lattice points on curves of genus 1, Proc. London Math. Soc. (2) 39 (1935), 431-466 and 40 (1935), 558.
• [M30]. K. Mahler, Über transzendente $P$-adische Zahlen, Compositio Math. 2 (1935), 259-275.
• [M47]. P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), 134-139.
• [M147]. D. J. Lewis and K. Mahler, On the representation of integers by binary forms, Acta Arithm. 6 (1961), 333-363.
• [M215]. K. Mahler, On Thue's theorem, Math. Scand. 55 (1984), 188-200.
• 1. A. Baker, Linear forms in the logarithms of algebraic numbers, Mathematika 13 (1966), 204-216.
• 2. A. Baker, Linear forms in the logarithms of algebraic numbers, II, Mathematika 14 (1967), 102-107.
• 3. A. Baker, Linear forms in the logarithms of algebraic numbers, IV, Mathematika 15 (1968), 204-216.
• 4. A. Baker, Contributions to the theory of Diophantine equations, Philos. Trans. Roy. Soc. London, Ser A 263 (1968), 173-208.
• 5. A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, Cambridge University Press, 2007.
• 6. M. A. Bean, An isoperimetric inequality for the area of plane regions defined by binary forms, Compos. Math. 92 (1994), 115-131.
• 7. M. A. Bennett, N. P. Dummigan, and T. D. Wooley, The representation of integers by binary additive forms, Compositio Mathematica 111 (1998), 15-33.
• 8. F. Beukers and H. P. Schlickewei, The equation $x+y=1$ in finitely generated groups, Acta Arith. 78 (1996), 189-199.
• 9. A. Bérczes, J.-H. Evertse, and K. Győry, Effective results for Diophantine equations over finitely generated domains, Acta Arith. 163 (2014), 71-100.
• 10. E. Bombieri and W. M. Schmidt, On Thue's equation, Invent. Math. 88 (1987), 69-81.
• 11. T. D. Browning, Equal sums of two $kth$ powers, J. Number Theory 96 (2002), 293-318.
• 12. Y. Bugeaud, Linear Forms in Logarithms and Applications, European Math. Soc. 2018.
• 13. Y. Bugeaud and J.-H. Evertse, On two notions of complexity of algebraic numbers, Acta Arith. 133 (2008), 221-250 (volume dedicated to Wolfgang Schmidt on the occasion of his 75th birthday).
• 14. Y. Bugeaud and K. Győry, Bounds for the solutions of Thue-Mahler equations and norm form equations, Acta Arith. 74 (1996), 273-292.
• 15. S. D. Chowla, Contributions to the analytic theory of numbers (II), J. Indian Math. Soc. 20 (1933) 121-128.
• 16. J. Coates, An effective $p$-adic analogue of a theorem of Thue, Acta Arith. 15 (1969), 279-305.
• 17. J. Coates, An effective $p$-adic analogue of a theorem of Thue II, The greatest prime factor of a binary form, Acta Arith. 16 (1970), 392-412.
• 18. H. Davenport and K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 160-167.
• 19. F. J. Dyson, The approximation of algebraic numbers by rationals, Acta Math. 79 (1947), 225-240.
• 20. P. Erdős, C. L. Stewart, and R. Tijdeman, Some diophantine equations with many solutions, Compositio Math. 66 (1988), 37-56.
• 21. J.-H. Evertse, Upper bounds for the numbers of solutions of Diophantine equations, PhD-thesis, Leiden, 1983, also published as MC-tract 168, Centrum voor Wiskunde en Informatica, Amsterdam, 1983.
• 22. J.-H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561-584.
• 23. J.-H. Evertse, On sums of $S$-units and linear recurrences, Compos. Math. 53 (1984), 225-244.
• 24. J.-H. Evertse, The number of solutions of the Thue-Mahler equation, J. Reine Angew. Math. 482 (1997), 121-149.
• 25. J.-H. Evertse and R. G. Ferretti, A further improvement of the Quantitative Subspace Theorem, Ann. Math. 177 (2013), 513-590.
• 26. J.-H. Evertse and K. Győry, Finiteness criteria for decomposable form equations, Acta Arith. 50 (1988), 357-379.
• 27. J.-H. Evertse and K. Győry, Effective results for unit equations over finitely generated domains, Math. Proc. Camb. Phil. Soc. 154 (2013), 351-380.
• 28. J.-H. Evertse and K. Győry, Unit Equations in Diophantine Number Theory, Cambridge University Press, 2015.
• 29. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366; Erratum, ibidum 75 (1984), 381.
• 30. A. O. Gel'fond, Transcendental and algebraic numbers, Dover, New York, 1960.
• 31. A. O. Gel'fond, Sur le septième problème de Hilbert, Izv. Akad. Nauk SSSR 7 (1934), 623-630.
• 32. A. O. Gel'fond, On approximating transcendental numbers by algebraic numbers, Dokl. Akad. Nauk SSSR 2 (1935), 177-182.
• 33. A. O. Gel'fond, Sur la divisibilité de la différence des puissances de deux nombres premiers par une puissance d'un idéal premier, Mat. Sbornik 7 (1940), 7-26.
• 34. F. Q. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, Journal of the American Mathematical Society, (1) 4 (1991), 793-805.
• 35. G. Greaves, Power-free values of binary forms, Quart. J. Math, (2) 43 (1992), 45-65.
• 36. G. Greaves, Representation of a number by the sum of two fourth powers, Mat. Zametki 55 (1994), 47-58.
• 37. K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné II, Publ. Math. Debrecen 21 (1974), 125-144.
• 38. K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné III, Publ. Math. Debrecen 23 (1976), 141-165.
• 39. K. Győry, On the number of solutions of linear equations in units of an algebraic number field, Comment. Math. Helv. 54 (1979), 583-600.
• 40. K. Győry, On $S$-integral solutions of norm form, discriminant form and index form equations, Studia Sci. Math. Hungar. 16 (1981), 149-161.
• 41. K. Győry and K. Yu, Bounds for the solutions of $S$-unit equations and decomposable form equations, Acta Arith. 123 (2006), 9-41.
• 42. D. R. Heath-Brown, The density of rational points on cubic surfaces, Acta Arith. 79 (1997), 17-30.
• 43. C. Hooley, On binary cubic forms, J. Reine Angew. Math. 226 (1967), 30-87.
• 44. C ~Hooley, On another sieve method and the numbers that are a sum of two $h^{th}$ powers, Proc. London Math. Soc. 43 (1981), 73-109.
• 45. C. Hooley, On binary quartic forms, J. Reine Angew. Math. 366 (1986), 32-52.
• 46. C. Hooley, On binary cubic forms: II, J. Reine Angew. Math. 521 (2000), 185-240.
• 47. C. Hooley, On totally reducible binary forms: II, Hardy-Ramanujan Journal 25 (2002), 22-49.
• 48. S. Konyagin and K. Soundararajan, Two S-unit equations with many solutions, J. Number Theory 124 (2007), 193-199.
• 49. S. Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27-43.
• 50. M. Laurent, Équations diophantines exponentielles, Invent. Math. 78 (1984), 299-327.
• 51. J. Liu, On $p$-adic Decomposable Form Inequalities, PhD-thesis, Leiden, 2015.
• 52. C. J. Parry, The $\mathfrak{p}$-adic generalisation of the Thue-Siegel theorem, Acta Math. 83 (1950), 1-100.
• 53. A. J. van der Poorten and H. P. Schlickewei, The growth condition for recurrence sequences, Macquarie Univ. Math. Rep. 82-0041 (1982).
• 54. J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C.R. Acad. Sci. Paris 305 (1987) 215-218.
• 55. D. Ridout, Rational approximations to algebraic numbers, Mathematika 4 (1957), 125-131.
• 56. K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1-20; corrigendum, 168.
• 57. H. P. Schlickewei, On norm form equations, J. Number Theory 9 (1977), 370-380.
• 58. H. P. Schlickewei, The quantitative subspace theorem for number fields, Compos. Math. 82 (1992), 245-273.
• 59. W. M. Schmidt, Linearformen mit algebraischen Koefficienten II, Math. Ann. 191 (1971), 1-20.
• 60. W. M. Schmidt, The subspace theorem in Diophantine approximations, Compos. Math. 69 (1989), 121-173.
• 61. T. Schneider, Transzendenzuntersuchungen periodischer Funktionen; I Transzendenz von Potenzen; II Transzendenzeigenschaften elliptischer Funktionen, J. Reine Angew. Math. 172 (1934), 65-74.
• 62. C. L. Siegel, Approximation algebraischer Zahlen, Math. Zeitschr. 10 (1921), 173-213.
• 63. C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuß. Akad. Wissensch. Phys.-math. Klasse $=$ Ges. Abh. Bd. I, Springer Verlag 1966, 209-266.
• 64. J. H. Silverman, Integer points on curves of genus 1, J. London Math. Soc. 28 (1983) 1-7.
• 65. C. Skinner and T. D. Wooley, Sums of two $k$-th powers, J. Reine Angew. Math. 462 (1995), 57-68.
• 66. C. L. Stewart, On the number of solutions to polynomial congruences and Thue equations, J. Amer. Math. Soc. (4) 4 (1991), 793-835.
• 67. C. L. Stewart, Cubic Thue equations with many solutions, Int. Math. Res. Not. IMRN (2008), no. 13, Art. ID rnn040, 11 pp.
• 68. C. L. Stewart, Integer points on cubic Thue equations, C. R. Math. Acad. Sci. Paris, Ser. I 347 (2009), 715-718.
• 69. C. L. Stewart and J. Top, On ranks of twists of elliptic curves and power-free values of binary forms, J. Amer. Math. Soc. (4) 8 (1995), 943-972.
• 70. C. L. Stewart and S. Y. Xiao, On the representation of integers by binary forms, Math. Ann. 375 (2019), no. 1-2, 133-163.
• 71. C. L. Stewart and S. Y. Xiao, On the representation of k-free integers by binary forms, arXiv:1612.00487.
• 72. A. Thue, Über Annäherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284-305.
• 73. J. L. Thunder, On cubic Thue inequalities and a result of Mahler, Acta Arith. 83 (1998), 31-44.
• 74. J. L. Thunder, Decomposable form inequalities, Ann. Math. 153 (2001), 767-804.
• 75. J. L. Thunder, Asymptotic estimates for the number of integer solutions to decomposable form inequalities, Compos. Math. 141 (2005), 271-292.
• 76. T. D. Wooley, Sums of two cubes, Int. Math. Res. Notices 4 (1995), 181-185.
• 77. U. Zannier (ed.), On Some Applications of Diophantine Approximations, Edizioni Della Normale, Pisa, 2014.

### Affiliation

Evertse, Jan-Hendrik
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands
Győry, Kálmán
Institute of Mathematics, University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary
Stewart, Cameron L.
Dept. of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada