Logarithms of algebraic numbers
Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 499-535.

Cet article est consacré à une nouvelle démonstration de la transcendance des évaluations du logarithme archimédien en tout nombre algébrique, exception faite de l’unité. Par analogie avec d’autres démonstrations de ce résultat, nous employons une variante de l’approximation de Padé pour le logarithme népérien. Une différence cependant : la construction de ces approximations de Padé est ici obtenue par le lemme de Siegel. La méthode exposée suggère des généralisations qui sont aussi évoquées.

This article is devoted to a new proof of transcendence for evaluations of the archimedean logarithm at all algebraic numbers except unity. As in other proofs of the same theorem, a sort of Padé approximation for the natural logarithm is employed. Whereas in previous approaches the used Padé approximants have been obtained rather ad hoc, we construct them here systematically by Siegel’s Lemma. The method presented suggests some generalizations, which are also briefly surveyed.

DOI : 10.5802/jtnb.912
Classification : 11J82
Kühne, Lars 1

1 Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa Italy
@article{JTNB_2015__27_2_499_0,
     author = {K\"uhne, Lars},
     title = {Logarithms of algebraic numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {499--535},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     doi = {10.5802/jtnb.912},
     mrnumber = {3393165},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.912/}
}
TY  - JOUR
AU  - Kühne, Lars
TI  - Logarithms of algebraic numbers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2015
SP  - 499
EP  - 535
VL  - 27
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.912/
DO  - 10.5802/jtnb.912
LA  - en
ID  - JTNB_2015__27_2_499_0
ER  - 
%0 Journal Article
%A Kühne, Lars
%T Logarithms of algebraic numbers
%J Journal de théorie des nombres de Bordeaux
%D 2015
%P 499-535
%V 27
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.912/
%R 10.5802/jtnb.912
%G en
%F JTNB_2015__27_2_499_0
Kühne, Lars. Logarithms of algebraic numbers. Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 499-535. doi : 10.5802/jtnb.912. http://www.numdam.org/articles/10.5802/jtnb.912/

[1] F. Amoroso and C. Viola, Approximation measures for logarithms of algebraic numbers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30, 1 (2001), 225–249. | Numdam | MR | Zbl

[2] Y. André, G-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, (1989). | MR | Zbl

[3] —, Théorie des motifs et interprétation géométrique des valeurs p-adiques de G-functions (une introduction), Number theory (Paris, 1992–1993), London Math. Soc. Lecture Note Ser., 215, Cambridge Univ. Press, Cambridge, (1995), 37–60.

[4] —, G-fonctions et transcendance, J. Reine Angew. Math. 476 (1996), 95–125. | MR

[5] —, Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. of Math. (2) 151, 2 (2000), 705–740. | MR

[6] —, Séries Gevrey de type arithmétique. II. Transcendance sans transcendance, Ann. of Math. (2) 151, 2 (2000), 741–756. | MR

[7] —, Arithmetic Gevrey series and transcendence. A survey, J. Théor. Nombres Bordeaux 15, 1 (2003), 1–10, Les XXIIèmes Journées Arithmetiques (Lille, 2001).

[8] Y. André and Francesco Baldassarri, Geometric theory of G-functions, Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, (1997), 1–22. | MR | Zbl

[9] A. Baker, Approximations to the logarithms of certain rational numbers, Acta Arith. 10 (1964), 315–323. | MR | Zbl

[10] A. Baker, Linear forms in the logarithms of algebraic numbers. I, II, III, Mathematika 13 (1966), 204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228. | MR | Zbl

[11] —, Logarithmic forms and the abc-conjecture, Number theory (Eger, 1996), de Gruyter, Berlin (1998), 37–44. | MR

[12] A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, New Mathematical Monographs, vol. 9, Cambridge University Press, Cambridge, (2007). | MR | Zbl

[13] D. Bertrand, On André’s proof of the Siegel-Shidlovsky theorem, Colloque Franco-Japonais: Théorie des Nombres Transcendants (Tokyo, 1998), Sem. Math. Sci., 27, Keio Univ., Yokohama, (1999), 51–63. | MR

[14] D. Bertrand and F. Beukers, Équations différentielles linéaires et majorations de multiplicités, Ann. Sci. École Norm. Sup. (4) 18, 1 (1985), 181–192. | Numdam | MR | Zbl

[15] A. A. Bolibrukh, The Fuchs inequality on a compact Kähler manifold, Dokl. Akad. Nauk 380, 4 (2001), 448–451. | MR | Zbl

[16] E. Bombieri, On G-functions, Recent progress in analytic number theory, 2 (Durham, 1979), Academic Press, London, (1981), 1–67. | MR | Zbl

[17] —, Effective Diophantine approximation on G m , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), no. 1, 61–89. | Numdam | MR

[18] E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73, 1 (1983), 11–32. | MR | Zbl

[19] Y. Bugeaud and M. Laurent, Minoration effective de la distance p-adique entre puissances de nombres algébriques, J. Number Theory 61, 2 (1996), 311–342. | MR | Zbl

[20] D.V. Chudnovsky and G.V. Chudnovsky, Applications of Padé approximations to Diophantine inequalities in values of G-functions, Number theory (New York, 1983–84), Lecture Notes in Math., 1135, Springer, Berlin, (1985), 9–51. | MR | Zbl

[21] P. Dèbes, G-fonctions et théorème d’irreductibilité de Hilbert, Acta Arith. 47, 4 (1986), 371–402. | MR | Zbl

[22] B. Dwork, G. Gerotto, and F.J. Sullivan, An introduction to G-functions, Annals of Mathematics Studies, 133, Princeton University Press, Princeton, NJ, (1994). | MR | Zbl

[23] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73, 3 (1983), 349–366. | MR | Zbl

[24] A.I. Galočkin, Lower bounds of linear forms of the values of certain G-functions, Mat. Zametki 18, 4 (1975), 541–552. | MR

[25] A. Gelfond, On Hilbert’s seventh problem, Mathematics of the USSR - Izvestiya 7 (1934), 623–630. | Zbl

[26] A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103. | Numdam | MR | Zbl

[27] C. Hermite, Sur la fonction exponentielle, C. R. Math. Acad. Sci. Paris 77 (1873), 18–24, 74–79, 226–233, 285–293.

[28] K. Mahler, On the approximation of logarithms of algebraic numbers, Philos. Trans. R. Soc. Lond., Ser. A 245 (1953), 371–398 (English). | MR | Zbl

[29] —, Lectures on transcendental numbers, Lecture Notes in Mathematics, 546, Springer-Verlag, Berlin, 1976. | MR

[30] R. Marcovecchio, The Rhin-Viola method for log2, Acta Arith. 139, 2 (2009), 147–184. | MR | Zbl

[31] R. Marcovecchio and C. Viola, Irrationality and nonquadraticity measures for logarithms of algebraic numbers, J. Aust. Math. Soc. 92, 2 (2012), 237–267. | MR | Zbl

[32] M. Mignotte, Approximations rationnelles de π et quelques autres nombres, Journées Arithmétiques (Grenoble, 1973), Soc. Math. France, Paris (1974) 121–132. Bull. Soc. Math. France, Mém. 37. | Numdam | MR | Zbl

[33] H.L. Montgomery and R.C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007. | MR | Zbl

[34] Y. Nesterenko and M. Waldschmidt, On the approximation of the values of the exponential function and logarithm by algebraic numbers, Diophantine approximations (Proceedings of papers dedicated to the memory of N. I. Fel’dman) (Yuri Nesterenko, ed.), Center for applied research under Mech.-Math. Faculty of MSU, Moscow, (1996).

[35] M.S. Nurmagomedov, The arithmetical properties of the values of G-functions, Vestnik Moskov. Univ. Ser. I Mat. Meh. 26, 6 (1971) 79–86. | MR | Zbl

[36] N. Ratazzi and E. Ullmo, Galois + équidistribution = Manin-Mumford, Arithmetic geometry, Clay Math. Proc., 8, Amer. Math. Soc., Providence, RI, (2009), 419–430. | MR | Zbl

[37] E. Reyssat, Mesures de transcendance pour les logarithmes de nombres rationnels, Diophantine approximations and transcendental numbers (Luminy, 1982), Progr. Math., 31, Birkhäuser Boston, Boston, MA, (1983), 235–245. | MR | Zbl

[38] T. Schneider, Transzendenzuntersuchungen periodischer Funktionen I, J. Reine Angew. Math. 172 (1934), 65–69. | Zbl

[39] A. B. Shidlovskii, Transcendental numbers, de Gruyter Studies in Mathematics, 12, Walter de Gruyter & Co., Berlin, (1989), Translated from the Russian by Neal Koblitz, With a foreword by W. Dale Brownawell. | MR | Zbl

[40] C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abhandlungen der Königlich-Preußischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse Nr. 1 (1929).

[41] C. L. Stewart and Kun Rui Yu, On the abc conjecture, Math. Ann. 291, 2 (1991), 225–230. | MR | Zbl

[42] C. L. Stewart and Kunrui Yu, On the abc conjecture. II, Duke Math. J. 108, 1 (2001), 169–181. | MR | Zbl

[43] M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, (2003). | MR | Zbl

[44] F. von Lindemann, Über die Zahl π, Math. Ann. 20 (1882), 213–225. | MR

[45] M. Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 326, Springer-Verlag, Berlin, (2000), Transcendence properties of the exponential function in several variables. | MR | Zbl

[46] M. Waldschmidt, On a problem of Mahler concerning the approximation of exponentials and logarithms, Publ. Math. Debrecen 56, 3-4 (2000), 713–738, Dedicated to Professor Kálmán Győry on the occasion of his 60th birthday. | MR | Zbl

[47] K. Weierstrass, Zu Lindemann’s Abhandlung ’über die Ludolph’sche Zahl’, Sitzungsberichte der Königlich-Preußischen Akademie der Wissenschaften (1885), 1067–1085.

[48] K. Yu, p-adic logarithmic forms and group varieties. I, J. Reine Angew. Math. 502 (1998), 29–92. | MR | Zbl

[49] —, p-adic logarithmic forms and group varieties. II, Acta Arith. 89, 4 (1999), 337–378. | MR | Zbl

[50] —, p-adic logarithmic forms and group varieties. III, Forum Math. 19, 2 (2007), 187–280. | MR | Zbl

[51] D. Zagier, The dilogarithm function, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, (2007), 3–65. | MR | Zbl

[52] U. Zannier (ed.), On Some Applications of Diophantine Approximationens (a translation of Carl Ludwig Siegel’s Über einige Anwendungen diophantischer Approximationen), Monographs, 2, Edizioni della Normale, Pisa, (2014).

Cité par Sources :