# Elliptic Functions and Transcendence

• Michel Waldschmidt
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 17)

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like π and π that are related to the classical exponential function, it turns out that elliptic functions are required (so far, this should not last forever!) to prove transcendence results and get a better understanding of the situation. First we briefly recall some of the basic transcendence results related to the exponential function (Section l). Next, in Section 2, we survey the main properties of elliptic functions that are involved in transcendence theory. We survey transcendence theory of values of elliptic functions in Section 3, linear independence in Section 4, and algebraic independence in Section 5. This splitting is somewhat artificial but convenient. Moreover, we restrict ourselves to elliptic functions, even when many results are only special cases of statements valid for abelian functions. A number of related topics are not considered here (e.g., heights, p-adic theory, theta functions, Diophantine geometry on elliptic curves).

Transcendental numbers elliptic functions elliptic curves elliptic integrals algebraic independence transcendence measures measures of algebraic independence Diophantine approximation.

## References

1. 1.
M. Ably ‘‘Résultats quantitatifs d’indépendance algébrique pour les groupes algébriques,’’ J. Number Theory 42 (1992), no. 2, pp. 194–231.
2. 2.
M. Ably ‘‘Formes linéaires de logarithmes de points algébriques sur une courbe elliptique de type CM,’’ Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, pp. 1–33.
3. 3.
M. Ably & É. Gaudron ‘‘Approximation diophantienne sur les courbes elliptiques ‘a multiplication complexe,’’ C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, pp. 629–634.
4. 4.
M. Abramowitz & I. A. Stegun – Handbook of mathematical functions with formulas, graphs, and mathematical tables, A Wiley-Interscience Publication, New York: John Wiley & Sons, Inc; Washington, D.C, 1984, Reprint of the 1972 ed.Google Scholar
5. 5.
M. Anderson – ‘‘Inhomogeneous linear forms in algebraic points of an elliptic function,’’ in Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977, pp. 121–143.Google Scholar
6. 6.
M. Anderson & D. W. Masser ‘‘Lower bounds for heights on elliptic curves,’’ Math. Z. 174 (1980), no. 1, pp. 23–34.
7. 7.
Y. André – G-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989.Google Scholar
8. 8.
Y. André ‘‘G-fonctions et transcendance,’’ J. reine angew. Math. 476 (1996), pp. 95–125.
9. 9.
Y. André – Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, vol. 17, Société Mathématique de France, Paris, 2004.Google Scholar
10. 10.
A. Baker ‘‘On the quasi-periods of the Weierstrass ζ -function,’’ Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), pp. 145–157.Google Scholar
11. 11.
A. Baker ‘‘An estimate for the ℘ -function at an algebraic point,’’ Amer. J. Math. 92 (1970), pp. 619–622.
12. 12.
A. Baker – ‘‘On the periods of the Weierstrass ℘ -function,’’ in Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 155–174.Google Scholar
13. 13.
A. Baker – Transcendental number theory, Cambridge University Press, London, 1975.
14. 14.
K. Barré ‘‘Mesures de transcendance pour l’invariant modulaire,’’ C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 5, pp. 447–452.
15. 15.
K. Barré ‘‘Mesure d’approximation simultanée de q et J(q),’’ J. Number Theory 66 (1997), no. 1, pp. 102–128.
16. 16.
K. Barré-Sirieix, G. Diaz, F. Gramain & G. Philibert ‘‘Une preuve de la conjecture de Mahler-Manin,’’ Invent. Math. 124 (1996), no. 1–3, pp. 1–9.
17. 17.
C. Bertolin ‘‘Périodes de 1-motifs et transcendance,’’ J. Number Theory 97 (2002), no. 2, pp. 204–221.
18. 18.
D. Bertrand ‘‘Séries d’Eisenstein et transcendance,’’ Bull. Soc. Math. France 104 (1976), no. 3, pp. 309–321.
19. 19.
D. Bertrand – ‘‘Transcendance de valeurs de la fonction gamma d’après G. V. Chudnovsky (Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698–701),’’ in Séminaire Delange-Pisot-Poitou, 17e année (1975/76), Théorie des nombres: Fasc. 2, Exp. No. G8, Secrétariat Math., Paris, 1977, p. 5.Google Scholar
20. 20.
D. Bertrand – ‘‘Fonctions modulaires, courbes de Tate et indépendance algébrique,’’ in Séminaire Delange-Pisot-Poitou, 19e année: 1977/78, Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1978, p. Exp. No. 36, 11.Google Scholar
21. 21.
D. Bertrand – ‘‘Modular function and algebraic independence,’’ in Proceedings of the Conference on p-adic Analysis (Nijmegen, 1978), Report, vol. 7806, Katholieke Univ. Nijmegen, 1978, pp. 16–23.Google Scholar
22. 22.
D. Bertrand – ‘‘Propriétés arithmétiques des dérivées de la fonction modulaire j(τ),’’ in Séminaire de Théorie des Nombres 1977–1978, CNRS, Talence, 1978, p. Exp. No. 22, 4.Google Scholar
23. 23.
D. Bertrand – ‘‘Fonctions modulaires et indépendance algébrique. II,’’ in Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), Astérisque, vol. 61, Soc. Math. France, Paris, 1979, pp. 29–34.Google Scholar
24. 24.
D. Bertrand ‘‘Sur les périodes de formes modulaires,’’ C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 10, pp. A531–A534.
25. 25.
D. Bertrand ‘‘Variétés abeliennes et formes linéaires d’intégrales elliptiques,’’ in Théorie des nombres, Sémin. Delange-Pisot-Poitou, Paris 1979–80, Prog. Math. 12, 15–27, 1981.Google Scholar
26. 26.
D. Bertrand – ‘‘Endomorphismes de groupes algébriques; applications arithmétiques,’’ in Diophantine approximations and transcendental numbers (Luminy, 1982), Progr. Math., vol. 31, Birkhäuser Boston, Boston, MA, 1983, pp. 1–45.Google Scholar
27. 27.
D. Bertrand ‘‘Theta functions and transcendence,’’ Ramanujan J. 1 (1997), no. 4, pp. 339–350, International Symposium on Number Theory (Madras, 1996).
28. 28.
D. Bertrand & Y. Z. Flicker ‘‘Linear forms on abelian varieties over local fields,’’ Acta Arith. 38 (1980/81), no. 1, pp. 47–61.
29. 29.
D. Bertrand & M. Laurent ‘‘Propriétés de transcendance de nombres liés aux fonctions thêta,’’ C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 16, pp. 747–749.
30. 30.
D. Bertrand & D. Masser ‘‘Formes linéaires d’intégrales abéliennes,’’ C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 16, pp. A725–A727.
31. 31.
D. Bertrand & D. Masser ‘‘Linear forms in elliptic integrals,’’ Invent. Math. 58 (1980), no. 3, pp. 283–288.
32. 32.
V. Bosser – ‘‘Indépendance algébrique de valeurs de séries d’Eisenstein (théorème de Nesterenko),’’ in Formes modulaires et transcendance, Sémin. Congr., vol. 12, Soc. Math. France, Paris, 2005, pp. 119–178.Google Scholar
33. 33.
J.-B. Bost ‘‘Périodes et isogénies des variétés abéliennes sur les corps de nombres (d’après D. Masser et G. Wüstholz),’’ Astérisque 237 (1996), no. 4, pp. 115–161, Séminaire Bourbaki, Vol. 1994/95, Exp. No. 795.
34. 34.
N. Brisebarre & G. Philibert ‘‘Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j,’’ Journal of the Ramanujan Mathe-matical Society 20 (2005), no. 4, pp. 255–282.
35. 35.
W. D. Brownawell – ‘‘On the development of Gelcprime fond’s method,’’ in Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 18–44.Google Scholar
36. 36.
W. D. Brownawell – ‘‘Large transcendence degree revisited. I. Exponential and non-CM cases,’’ in Diophantine approximation and transcendence theory (Bonn, 1985), Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 149–173.Google Scholar
37. 37.
W. D. Brownawell & K. K. Kubota ‘‘The algebraic independence of Weierstrass functions and some related numbers,’’ Acta Arith. 33 (1977), no. 2, pp. 111–149.
38. 38.
W. D. Brownawell & R. Tubbs – ‘‘Large transcendence degree revisited. II. The CM case,’’ in Diophantine approximation and transcendence theory (Bonn, 1985), Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, pp. 175–188.
39. 39.
S. Bruiltet ‘‘D’une mesure d’approximation simultanée à une mesure d’irrationalité: le cas de Γ(1/4) et Γ(1/3),’’ Acta Arith. 104 (2002), no. 3, pp. 243–281.
40. 40.
P. Bundschuh ‘‘Ein Approximationsmass für transzendente Lösungen gewisser transzendenter Gleichungen,’’ J. reine angew. Math. 251 (1971), pp. 32–53.
41. 41.
P. Bundschuh ‘‘Zwei Bemerkungen über transzendente Zahlen,’’ Monatsh. Math. 88 (1979), no. 4, pp. 293–304.
42. 42.
E. B. Burger & R. Tubbs – Making transcendence transparent, Springer-Verlag, New York, 2004, An intuitive approach to classical transcendental number theory.
43. 43.
K. Chandrasekharan – Elliptic functions, Grundlehren der Mathematischen Wissenschaften, vol. 281, Springer-Verlag, Berlin, 1985.Google Scholar
44. 44.
S. Chowla & A. Selberg ‘‘On Epstein’s zeta-function,’’ J. reine angew. Math. 227 (1967), pp. 86–110.
45. 45.
G. V. Chudnovsky ‘‘Algebraic independence of constants connected with the exponential and the elliptic functions,’’ Dokl. Akad. Nauk Ukrain. SSR Ser. A 8 (1976), pp. 698–701, 767.Google Scholar
46. 46.
G. V. Chudnovsky ‘‘Indépendance algébrique des valeurs d’une fonction elliptique en des points algébriques. Formulation des résultats,’’ C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 8, pp. A439–A440.
47. 47.
G. V. Chudnovsky ‘‘Algebraic independence of the values of elliptic function at algebraic points,’’ Invent. Math. 61 (1980), no. 3, pp. 267–290, Elliptic analogue of the Lindemann-Weierstrass theorem.
48. 48.
G. V. Chudnovsky – ‘‘Algebraic independence of values of exponential and elliptic functions,’’ in Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (Helsinki), Acad. Sci. Fennica, 1980, pp. 339–350.Google Scholar
49. 49.
G. V. Chudnovsky ‘‘Indépendance algébrique dans la méthode de Gelfond-Schneider,’’ C. R. Acad. Sci., Paris, Sér. A 291 (1980), pp. 365–368, see Zbl 0456.10016.
50. 50.
G. V. Chudnovsky – Contributions to the theory of transcendental numbers, Mathematical Surveys and Monographs, vol. 19, American Mathematical Society, Providence, RI, 1984.Google Scholar
51. 51.
J. Coates ‘‘An application of the division theory of elliptic functions to Diophantine approximation,’’ Invent. Math. 11 (1970), pp. 167–182.
52. 52.
J. Coates ‘‘An application of the Thue-Siegel-Roth theorem to elliptic functions,’’ Proc. Cambridge Philos. Soc. 69 (1971), pp. 157–161.
53. 53.
J. Coates ‘‘Linear forms in the periods of the exponential and elliptic functions,’’ Invent. Math. 12 (1971), pp. 290–299.
54. 54.
J. Coates ‘‘The transcendence of linear forms in ω1, ω 2, η12, 2π i,’’ Amer. J. Math. 93 (1971), pp. 385–397.
55. 55.
J. Coates – ‘‘Linear relations between 2π i and the periods of two elliptic curves,’’ in Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972), Academic Press, New York, 1973, pp. 77–99.Google Scholar
56. 56.
J. Coates & S. Lang ‘‘Diophantine approximation on Abelian varieties with complex multiplication,’’ Invent. Math. 34 (1976), no. 2, pp. 129–133 (= [130] pp. 236–240).
57. 57.
H. Cohen – ‘‘Elliptic curves,’’ in From Number Theory to Physics (Les Houches, 1989), Springer, Berlin, 1992, pp. 212–237.Google Scholar
58. 58.
P. B. Cohen ‘‘On the coefficients of the transformation polynomials for the elliptic modular function,’’ Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 3, pp. 389–402.
59. 59.
P. B. Cohen ‘‘Perspectives de l’approximation diophantienne et de la transcendance,’’ Ramanujan J. 7 (2003), no. 1–3, pp. 367–384, Rankin memorial issues.
60. 60.
S. David ‘‘Minorations de formes linéaires de logarithmes elliptiques,’’ Mém. Soc. Math. France (N.S.) 62 (1995), pp. 1–143.
61. 61.
S. David – ‘‘On the height of subvarieties of group varieties,’’ (preprint).Google Scholar
62. 62.
S. David & N. Hirata-Kohno – ‘‘Recent progress on linear forms in elliptic logarithms,’’ in A panorama of number theory or the view from Baker’s garden (Zürich, 1999), Cambridge Univ. Press, Cambridge, 2002, pp. 26–37.Google Scholar
63. 63.
G. Diaz ‘‘Grands degrés de transcendance pour des familles d’exponentielles,’’ J. Number Theory 31 (1989), no. 1, pp. 1–23.
64. 64.
G. Diaz ‘‘Minorations de combinaisons linéaires non homogènes pour un logarithme elliptique,’’ C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 10, pp. 879–883.
65. 65.
G. Diaz ‘‘La conjecture des quatre exponentielles et les conjectures de D. Bertrand sur la fonction modulaire,’’ J. Théor. Nombres Bordeaux 9 (1997), no. 1, pp. 229–245.
66. 66.
G. Diaz ‘‘Transcendance et indépendance algébrique: liens entre les points de vue elliptique et modulaire,’’ Ramanujan J. 4 (2000), no. 2, pp. 157–199.
67. 67.
G. Diaz & G. Philibert ‘‘Growth properties of the modular function j,’’ J. Math. Anal. Appl. 139 (1989), no. 2, pp. 382–389.
68. 68.
D. Duverney, K. Nishioka, K. Nishioka & I. Shiokawa ‘‘Transcendence of Jacobi’s theta series,’’ Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 9,break pp. 202–203.
69. 69.
D. Duverney, K. Nishioka, K. Nishioka & I. Shiokawa ‘‘Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers,’’ Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 7, pp. 140–142.
70. 70.
D. Duverney, K. Nishioka, K. Nishioka & I. Shiokawa – ‘‘Transcendence of Jacobi’s theta series and related results,’’ in Number theory (Eger, 1996), W. de Gruyter, Berlin, 1998, pp. 157–168.Google Scholar
71. 71.
D. Duverney, K. Nishioka, K. Nishioka & I. Shiokawa ‘‘Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers,’’ Sūrikaisekikenkyūsho Kōkyūroku 1060 (1998), pp. 91–100, Number theory and its applications (Japanese) (Kyoto, 1997).
72. 72.
D. Duverney & I. Shiokawa – ‘‘On some arithmetical properties of Rogers-Ramanujan continued fraction,’’ in Colloque Franco-Japonais: Théorie des Nombres Transcendants (Tokyo, 1998), Sem. Math. Sci., vol. 27, Keio Univ., Yokohama, 1999, pp. 91–100.Google Scholar
73. 73.
D. Duverney & I. Shiokawa ‘‘On some arithmetical properties of Rogers-Ramanujan continued fraction,’’ Osaka J. Math. 37 (2000), no. 3, pp. 759–771.
74. 74.
A. Faisant & G. Philibert – ‘‘Mesure d’approximation simultanée pour la fonction modulaire j et résultats connexes,’’ in Séminaire de théorie des nombres, Paris 1984–85, Progr. Math., vol. 63, Birkhäuser Boston, Boston, MA, 1986, pp. 67–78.Google Scholar
75. 75.
A. Faisant & G. Philibert ‘‘Quelques résultats de transcendance liés à l’invariant modulaire j,’’ J. Number Theory 25 (1987), no. 2, pp. 184–200.
76. 76.
N. I. Fel′dman ‘‘On the measure of transcendency of the logarithms of algebraic numbers and elliptic constants,’’ Uspehi Matem. Nauk (N.S.) 4 (1949), no. 1(29), p. 190.Google Scholar
77. 77.
N. I. Fel′dman ‘‘The approximation of certain transcendental numbers. II. The approximation of certain numbers connected with the Weierstrass function ℘(z),’’ Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), pp. 153–176.
78. 78.
N. I. Fel′dman ‘‘Joint approximations of the periods of an elliptic function by algebraic numbers,’’ Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), pp. 563–576.
79. 79.
N. I. Fel′dman ‘‘An elliptic analog of an inequality of A. O. Gelcprime fond,’’ Trudy Moskov. Mat. Obšš. 18 (1968), pp. 65–76.Google Scholar
80. 80.
N. I. Fel′dman ‘‘The periods of elliptic functions,’’ Acta Arith. 24 (1973/74), pp. 477–489, Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, V.
81. 81.
N. I. Fel′dman ‘‘The algebraic independence of certain numbers,’’ Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1980), no. 4, pp. 46–50, 100.Google Scholar
82. 82.
N. I. Fel′dman ‘‘Algebraic independence of some numbers. II,’’ Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 25 (1982), pp. 109–123.
83. 83.
N. I. Fel′dman & Y. V. Nesterenko – ‘‘Transcendental numbers,’’ in Number theory, IV, Encyclopaedia Math. Sci., vol. 44, Springer, Berlin, 1998, pp. 1–345.Google Scholar
84. 84.
Y. Z. Flicker ‘‘Linear forms on arithmetic abelian varieties: ineffective bounds,’’ Mém. Soc. Math. France (N.S.) 2 (1980/81), pp. 41–47, Abelian functions and transcendental numbers (Colloq., ’Ecole Polytech., Palaiseau, 1979).
85. 85.
R. Franklin ‘‘The transcendence of linear forms in ω1, ω2, η1, η2, 2π i, log γ,’’ Acta Arith. 26 (1974/75), pp. 197–206.
86. 86.
É. Gaudron ‘‘Étude du cas rationnel de la théorie des formes linéaires de logarithmes,’’ J. Number Theory, 127 (2007). no. 2, pp. 220–261.
87. 87.
É. Gaudron ‘‘Formes linéaires de logarithmes effectives sur les variétés abéliennes,’’ submitted, 2004. Ann. Sci. de l’École Norm. Sup., 39 (2006), no. 5, pp. 699–773.
88. 88.
É. Gaudron ‘‘Mesures d’indépendance linéaire de logarithmes dans un groupe algébrique commutatif,’’ Invent. Math. 162 (2005), no. 1, pp. 137–188.
89. 89.
A. O. Gel′fond – Transcendental and algebraic numbers, Translated from the first Russian edition by Leo F. Boron, Dover Publications Inc., New York, 1960.Google Scholar
90. 90.
P. Graftieaux – ‘‘Théorème stéphanois et méthode des pentes,’’ in Formes modulaires et transcendance, Sémin. Congr., vol. 12, Soc. Math. France, Paris, 2005, pp. 179–213.Google Scholar
91. 91.
F. Gramain – ‘‘Quelques résultats d’indépendance algébrique,’’ in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin), 1998, pp. 173–182.Google Scholar
92. 92.
F. Gramain ‘‘Transcendance et fonctions modulaires,’’ J. Théor. Nombres Bordeaux 11 (1999), no. 1, pp. 73–90, Les XXèmes Journées Arithmétiques (Limoges, 1997).
93. 93.
P. Grinspan ‘‘A measure of simultaneous approximation for quasi-modular functions,’’ Ramanujan J. 5 (2001), no. 1, pp. 21–45.
94. 94.
P. Grinspan ‘‘Measures of simultaneous approximation for quasi-periods of abelian varieties,’’ J. Number Theory 94 (2002), no. 1, pp. 136–176.
95. 95.
B. H. Gross ‘‘On the periods of abelian integrals and a formula of Chowla and Selberg,’’ Invent. Math. 45 (1978), no. 2, pp. 193–211, With an appendix by David E. Rohrlich.
96. 96.
B. H. Gross ‘‘On an identity of Chowla and Selberg,’’ J. Number Theory 11 (1979), no. 3 S. Chowla Anniversary Issue, pp. 344–348.Google Scholar
97. 97.
T. Harase ‘‘The transcendence of eαω+β(2π i),’’ J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), pp. 279–285.
98. 98.
T. Harase ‘‘On the linear form of transcendental numbers; α1ω+α3 log α4,’’ J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, pp. 435–452.
99. 99.
M. Hindry & J. H. Silverman – Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An introduction.Google Scholar
100. 100.
N. Hirata-Kohno – ‘‘Formes linéaires d’intégrales elliptiques,’’ in Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 117–140.Google Scholar
101. 101.
N. Hirata-Kohno ‘‘Mesures de transcendance pour les quotients de périodes d’intégrales elliptiques,’’ Acta Arith. 56 (1990), no. 2, pp. 111–133.
102. 102.
N. Hirata-Kohno ‘‘Formes linéaires de logarithmes de points algébriques sur les groupes algébriques,’’ Invent. Math. 104 (1991), no. 2, pp. 401–433.
103. 103.
N. Hirata-Kohno – ‘‘Nouvelles mesures de transcendance liées aux groupes algébriques commutatifs,’’ in Approximations diophantiennes et nombres transcendants (Luminy, 1990), W. de Gruyter, Berlin, 1992, pp. 165–172.Google Scholar
104. 104.
N. Hirata-Kohno ‘‘Approximations simultanées sur les groupes algébriques commutatifs,’’ Compositio Math. 86 (1993), no. 1, pp. 69–96.
105. 105.
E. M. Jabbouri ‘‘Mesures d’indépendance algébrique de valeurs de fonctions elliptiques et abéliennes,’’ C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 9, pp. 375–378.
106. 106.
Y. M. Kholyavka – ‘‘Simultaneous approximations of the invariants of an elliptic function by algebraic numbers,’’ in Diophantine approximations, Part II (Russian), Moskov. Gos. Univ., Moscow, 1986, pp. 114–121.Google Scholar
107. 107.
Y. M. Kholyavka ‘‘Approximation of numbers that are connected with elliptic functions,’’ Mat. Zametki 47 (1990), no. 6, pp. 110–118, 160.Google Scholar
108. 108.
Y. M. Kholyavka ‘‘Approximation of the invariants of an elliptic function,’’ Ukrain. Mat. Zh. 42 (1990), no. 5, pp. 681–685.
109. 109.
Y. M. Kholyavka ‘‘On the approximation of some numbers related to ℘(z),’’ Vīsnik L′vīv. Unīv. Ser. Mekh. Mat. 34 (1990), pp. 88–89.
110. 110.
Y. M. Kholyavka ‘‘On the approximation of numbers connected with Weierstrass elliptic functions,’’ Sibirsk. Mat. Zh. 32 (1991), no. 1, pp. 212–216, 224.Google Scholar
111. 111.
Y. M. Kholyavka ‘‘Simultaneous approximation of numbers connected with elliptic functions,’’ Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1991), pp. 70–73.
112. 112.
Y. M. Kholyavka ‘‘Zeros of polynomials of Jacobi elliptic functions,’’ Ukraïn. Mat. Zh. 44 (1992), no. 11, p. 1624.
113. 113.
Y. M. Kholyavka – ‘‘On the approximation of certain numbers associated with Jacobi elliptic functions,’’ in Mathematical investigations (Ukrainian), Pr. L′ vīv. Mat. Tov., vol. 2, L′ vīv. Mat. Tov., L′ viv, 1993, pp. 10–13, 106.Google Scholar
114. 114.
Y. M. Kholyavka ‘‘On the approximation of numbers connected with ℘(z),’’ Vīsnik L′vīv. Unīv. Ser. Mekh. Mat. 38 (1993), p. 64.
115. 115.
N. Koblitz ‘‘Gamma function identities and elliptic differentials on Fermat curves,’’ Duke Math. J. 45 (1978), no. 1, pp. 87–99.
116. 116.
N. Koblitz – Introduction to elliptic curves and modular forms, second ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993.Google Scholar
117. 117.
N. Koblitz & D. Rohrlich ‘‘Simple factors in the Jacobian of a Fermat curve,’’ Canad. J. Math. 30 (1978), no. 6, pp. 1183–1205.
118. 118.
M. Kontsevich & D. Zagier – ‘‘Periods,’’ in Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808.Google Scholar
119. 119.
S. Lang ‘‘Diophantine approximations on toruses,’’ Amer. J. Math. 86 (1964), pp. 521–533 (= [129] pp. 313–325).
120. 120.
S. Lang – Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966 (= [129] pp. 396–506).
121. 121.
S. Lang ‘‘Transcendental numbers and Diophantine approximations,’’ Bull. Amer. Math. Soc. 77 (1971), pp. 635–677 (= [130] pp. 1–43).
122. 122.
S. Lang ‘‘Higher-dimensional Diophantine problems,’’ Bull. Amer. Math. Soc. 80 (1974), pp. 779–787 (= [130] pp. 102–110).
123. 123.
S. Lang ‘‘Diophantine approximation on abelian varieties with complex multiplication,’’ Advances in Math. 17 (1975), no. 3, pp. 281–336 (= [130] pp. 113–168).
124. 124.
S. Lang – Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, vol. 231, Springer-Verlag, Berlin, 1978.Google Scholar
125. 125.
S. Lang – ‘‘Relations de distributions et exemples classiques,’’ in Séminaire Delange-Pisot-Poitou, 19e année: 1977/78, Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1978, p. Exp. No. 40, 6 (= [131] pp. 59–65).Google Scholar
126. 126.
S. Lang – ‘‘Conjectured Diophantine estimates on elliptic curves,’’ in Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171 (= [131] pp. 212–228).Google Scholar
127. 127.
S. Lang – Elliptic functions, second ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987, With an appendix by J. Tate.Google Scholar
128. 128.
S. Lang – Cyclotomic fields I and II, second ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990, With an appendix by Karl Rubin.Google Scholar
129. 129.
S. Lang – Collected papers. Vol. I, Springer-Verlag, New York, 2000, 1952–1970.Google Scholar
130. 130.
S. Lang – Collected papers. Vol. II, Springer-Verlag, New York, 2000, 1971–1977.Google Scholar
131. 131.
S. Lang – Collected papers. Vol. III, Springer-Verlag, New York, 2000, 1978–1990.Google Scholar
132. 132.
M. Laurent ‘‘Transcendance de périodes d’intégrales elliptiques,’’ C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 15, pp. 699–701.
133. 133.
M. Laurent ‘‘Approximation diophantienne de valeurs de la fonction Beta aux points rationnels,’’ Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 1, pp. 53–65.
134. 134.
M. Laurent ‘‘Indépendance linéaire de valeurs de fonctions doublement quasi-périodiques,’’ C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 9, pp. A397–A399.
135. 135.
M. Laurent ‘‘Transcendance de périodes d’intégrales elliptiques,’’ J. reine angew. Math. 316 (1980), pp. 122–139.
136. 136.
M. Laurent – ‘‘Transcendance de périodes d’intégrales elliptiques,’’ in Séminaire Delange-Pisot-Poitou, 20e année: 1978/1979. Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1980, p. Exp. No. 13, 4.Google Scholar
137. 137.
M. Laurent ‘‘Transcendance de périodes d’intégrales elliptiques. II,’’ J. reine angew. Math. 333 (1982), pp. 144–161.
138. 138.
K. Mahler ‘‘On algebraic differential equations satisfied by automorphic functions,’’ J. Austral. Math. Soc. 10 (1969), pp. 445–450.
139. 139.
K. Mahler ‘‘Remarks on a paper by W. Schwarz,’’ J. Number Theory 1 (1969), pp. 512–521.
140. 140.
K. Mahler ‘‘On the coefficients of the 2n-th transformation polynomial for jω,’’ Acta Arith. 21 (1972), pp. 89–97.
141. 141.
K. Mahler ‘‘On the coefficients of transformation polynomials for the modular function,’’ Bull. Austral. Math. Soc. 10 (1974), pp. 197–218.
142. 142.
Y. Manin ‘‘Cyclotomic fields and modular curves,’’ Uspekhi Mat. Nauk 26 (1971), no. 6, pp. 7–71, Engl. Transl. Russ. Math. Surv. 26 (1971), no. 6, pp. 7–78.
143. 143.
D. W. Masser ‘‘On the periods of the exponential and elliptic functions,’’ Proc. Cambridge Philos. Soc. 73 (1973), pp. 339–350.
144. 144.
D. W. Masser – Elliptic functions and transcendence, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 437.
145. 145.
D. W. Masser ‘‘Linear forms in algebraic points of Abelian functions. I,’’ Math. Proc. Cambridge Philos. Soc. 77 (1975), pp. 499–513.
146. 146.
D. W. Masser ‘‘On the periods of Abelian functions in two variables,’’ Mathematika 22 (1975), no. 2, pp. 97–107.
147. 147.
D. W. Masser ‘‘Sur les points algébriques d’une variété abélienne,’’ C. R. Acad. Sci. Paris Sér. A-B 280 (1975), pp. A11–A12.
148. 148.
D. W. Masser – ‘‘Transcendence and abelian functions,’’ in Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Soc. Math. France, Paris, 1975, pp. 177–182. Astérisque, Nos. 24–25.Google Scholar
149. 149.
D. W. Masser ‘‘Linear forms in algebraic points of Abelian functions. II,’’ Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, pp. 55–70.
150. 150.
D. W. Masser ‘‘Linear forms in algebraic points of Abelian functions. III,’’ Proc. London Math. Soc. (3) 33 (1976), no. 3, pp. 549–564.
151. 151.
D. W. Masser ‘‘A note on a paper of Richard Franklin: [85],’’ Acta Arith. 31 (1976), no. 2, pp. 143–152.
152. 152.
D. W. Masser ‘‘Division fields of elliptic functions,’’ Bull. London Math. Soc. 9 (1977), no. 1, pp. 49–53.
153. 153.
D. W. Masser – ‘‘A note on Abelian functions,’’ in Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977, pp. 145–147.Google Scholar
154. 154.
D. W. Masser – ‘‘Some vector spaces associated with two elliptic functions,’’ in Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), Academic Press, London, 1977, pp. 101–119.Google Scholar
155. 155.
D. W. Masser ‘‘The transcendence of certain quasi-periods associated with Abelian functions in two variables,’’ Compositio Math. 35 (1977), no. 3, pp. 239–258.
156. 156.
D. W. Masser – ‘‘The transcendence of definite integrals of algebraic functions,’’ in Journées Arithmétiques de Caen (Univ. Caen, 1976), Soc. Math. France, Paris, 1977, pp. 231–238. Astérisque No. 41–42.Google Scholar
157. 157.
D. W. Masser ‘‘Diophantine approximation and lattices with complex multiplication,’’ Invent. Math. 45 (1978), no. 1, pp. 61–82.
158. 158.
D. W. Masser – ‘‘Heights, transcendence, and linear independence on commutative group varieties,’’ in Diophantine approximation (Cetraro, 2000), Lecture Notes in Math., vol. 1819, Springer, Berlin, 2003, pp. 1–51.Google Scholar
159. 159.
D. W. Masser ‘‘Sharp estimates for Weierstrass elliptic functions,’’ J. Anal. Math. 90 (2003), pp. 257–302.
160. 160.
D. W. Masser & G. Wüstholz ‘‘Algebraic independence properties of values of elliptic functions,’’ in Journées arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 360–363, 1982.Google Scholar
161. 161.
D. W. Masser & G. Wüstholz ‘‘Fields of large transcendence degree generated by values of elliptic functions,’’ Invent. Math. 72 (1983), no. 3, pp. 407–464.
162. 162.
D. W. Masser & G. Wüstholz ‘‘Algebraic independence of values of elliptic functions,’’ Math. Ann. 276 (1986), no. 1, pp. 1–17.
163. 163.
D. W. Masser & G. Wüstholz ‘‘Estimating isogenies on elliptic curves,’’ Invent. Math. 100 (1990), no. 1, pp. 1–24.
164. 164.
D. W. Masser & G. Wüstholz ‘‘Galois properties of division fields of elliptic curves,’’ Bull. London Math. Soc. 25 (1993), no. 3, pp. 247–254.
165. 165.
N. D. Nagaev ‘‘Approximation of the transcendental quotient of two algebraic points of the function ℘ (z) with complex multiplication,’’ Mat. Zametki 20 (1976), no. 1, pp. 47–60.
166. 166.
J. V. Nesterenko – ‘‘On the algebraical independence of algebraic numbers to algebraic powers,’’ in Diophantine approximations and transcendental numbers (Luminy, 1982), Progr. Math., vol. 31, Birkhäuser Boston, Mass., 1983, pp. 199–220.Google Scholar
167. 167.
Y. V. Nesterenko ‘‘Algebraic independence of algebraic powers of algebraic numbers,’’ Mat. Sb. (N.S.) 123(165) (1984), no. 4, pp. 435–459.Google Scholar
168. 168.
Y. V. Nesterenko ‘‘Degrees of transcendence of some fields that are generated by values of an exponential function,’’ Mat. Zametki 46 (1989), no. 3, pp. 40–49, 127.Google Scholar
169. 169.
Y. V. Nesterenko ‘‘Measure of algebraic independence of values of an elliptic function at algebraic points,’’ Uspekhi Mat. Nauk 40 (1985), no. 4(244), pp. 221–222.Google Scholar
170. 170.
Y. V. Nesterenko – ‘‘On a measure of algebraic independence of the values of elliptic functions,’’ in Approximations diophantiennes et nombres transcendants (Luminy, 1990), W. de Gruyter, Berlin, 1992, pp. 239–248.Google Scholar
171. 171.
Y. V. Nesterenko ‘‘On the measure of algebraic independence of values of an elliptic function,’’ Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 4, pp. 155–178.
172. 172.
Y. V. Nesterenko ‘‘Modular functions and transcendence problems,’’ C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 10, pp. 909–914.
173. 173.
Y. V. Nesterenko ‘‘Modular functions and transcendence questions,’’ Mat. Sb. 187 (1996), no. 9, pp. 65–96.
174. 174.
Y. V. Nesterenko ‘‘On the measure of algebraic independence of values of Ramanujan functions,’’ Tr. Mat. Inst. Steklova 218 (1997), no. Anal. Teor. Chisel i Prilozh., pp. 299–334.
175. 175.
Y. V. Nesterenko – ‘‘Algebraic independence of π and eπ,’’ in Number theory and its applications (Ankara, 1996), Lecture Notes in Pure and Appl. Math., vol. 204, Dekker, New York, 1999, pp. 121–149.Google Scholar
176. 176.
Y. V. Nesterenko ‘‘On the algebraic independence of values of Ramanujan functions,’’ Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (2001), pp. 6–10, 70.Google Scholar
177. 177.
Y. V. Nesterenko & P. Philippon – Introduction to algebraic independence theory, Lecture Notes in Mathematics, vol. 1752, Springer-Verlag, Berlin, 2001, With contributions from F. Amoroso, D. Bertrand, W. D. Brownawell, G. Diaz, M. Laurent, Yuri V. Nesterenko, K. Nishioka, Patrice Philippon, G. Rémond, D. Roy and M. Waldschmidt, edited by Nesterenko and Philippon.Google Scholar
178. 178.
F. Pellarin ‘‘The isogeny theorem and the irreducibility theorem for elliptic curves: a survey,’’ Rend. Sem. Mat. Univ. Politec. Torino 53 (1995), no. 4, pp. 389–404, Number theory, II (Rome, 1995).
179. 179.
F. Pellarin ‘‘The Ramanujan property and some of its connections with Diophantine geometry,’’ Riv. Mat. Univ. Parma (7) 3* (2004), pp. 275–288.
180. 180.
F. Pellarin – ‘‘Introduction aux formes modulaires de Hilbert et à leurs propriétés différentielles,’’ in Formes modulaires et transcendance, Sémin. Congr., vol. 12, Soc. Math. France, Paris, 2005, pp. 215–269.Google Scholar
181. 181.
G. Philibert ‘‘Une mesure d’indépendance algébrique,’’ Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, pp. 85–103.
182. 182.
P. Philippon – ‘‘Indépendance algébrique de valeurs de fonctions elliptiques p-adiques,’’ in Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979) (Kingston, Ont.), Queen’s Papers in Pure and Appl. Math., vol. 54, Queen’s Univ., 1980, pp. 223–235.Google Scholar
183. 183.
P. Philippon ‘‘Indépendance algébrique et variétés abéliennes,’’ C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 7, pp. 257–259.
184. 184.
P. Philippon ‘‘Variétés abéliennes et indépendance algébrique. I,’’ Invent. Math. 70 (1982/83), no. 3, pp. 289–318.
185. 185.
P. Philippon ‘‘Variétés abéliennes et indépendance algébrique. II. Un analogue abélien du théorème de Lindemann-Weierstrass,’’ Invent. Math. 72 (1983), no. 3, pp. 389–405.
186. 186.
P. Philippon ‘‘Une approche méthodique pour la transcendance et l’indépendance algébrique de valeurs de fonctions analytiques,’’ J. Number Theory 64 (1997), no. 2, pp. 291–338.
187. 187.
P. Philippon ‘‘Indépendance algébrique et K-fonctions,’’ J. reine angew. Math. 497 (1998), pp. 1–15.
188. 188.
P. Philippon & M. Waldschmidt ‘‘Formes linéaires de logarithmes sur les groupes algébriques commutatifs,’’ Illinois J. Math. 32 (1988), no. 2, pp. 281–314.
189. 189.
P. Philippon & M. Waldschmidt – ‘‘Formes linéaires de logarithmes elliptiques et mesures de transcendance,’’ in Théorie des nombres (Québec, PQ, 1987), W. de Gruyter, Berlin, 1989, pp. 798–805.Google Scholar
190. 190.
J. Popken & K. Mahler ‘‘Ein neues Prinzip Transzendenzbeweise,’’ Proc. Akad. Wet. Amsterdam 38 (1935), pp. 864–871.
191. 191.
K. Ramachandra ‘‘Contributions to the theory of transcendental numbers. I, II,’’ Acta Arith. 14 (1967/68), 65–72; ibid. 14 (1967/1968), pp. 73–88.Google Scholar
192. 192.
K. Ramachandra – Lectures on transcendental numbers, The Ramanujan Institute Lecture Notes, vol. 1, The Ramanujan Institute, Madras, 1969.Google Scholar
193. 193.
S. Ramanujan ‘‘On certain arithmetical functions,’’ Trans. Camb. Phil. Soc. 22 (1916), pp. 159–184, Collected Papers of Srinivasa Ramanujan, Chelsea Publ., N.Y. 1927, Nˆ18, 136–162.Google Scholar
194. 194.
G. Rémond & F. Urfels ‘‘Approximation diophantienne de logarithmes elliptiques p-adiques,’’ J. Number Theory 57 (1996), no. 1, pp. 133–169.
195. 195.
É. Reyssat – ‘‘Mesures de transcendance de nombres liés aux fonctions elliptiques,’’ in Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1977, p. Exp. No. G22, 3.Google Scholar
196. 196.
É. Reyssat ‘‘Mesures de transcendance de nombres liés aux fonctions exponentielles et elliptiques,’’ C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, pp. A977–A980.
197. 197.
É. Reyssat – ‘‘Travaux récents de G. V. Chudnovsky,’’ in Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1977, p. Exp. No. 29, 7.Google Scholar
198. 198.
É. Reyssat ‘‘Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielle,’’ Bull. Soc. Math. France 108 (1980), no. 1, pp. 47–79.
199. 199.
É. Reyssat ‘‘Approximation de nombres liés à la fonction sigma de Weierstrass,’’ Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 1, pp. 79–91.
200. 200.
É. Reyssat ‘‘Fonctions de Weierstrass et indépendance algébrique,’’ C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 10, pp. A439–A441.
201. 201.
É. Reyssat – ‘‘Propriétés d’indépendance algébrique de nombres liés aux fonctions de Weierstrass,’’ Acta Arith. 41 (1982), no. 3, pp. 291–310.
202. 202.
N. Saradha ‘‘Transcendence measure for η/ω,’’ Acta Arith. 92 (2000), no. 1, pp. 11–25.
203. 203.
T. Schneider ‘‘Transzendenzuntersuchungen periodischer Funktionen. II. Transzendenzeigenschaften elliptischer Funktionen,’’ J. reine angew. Math. 172 (1934), pp. 70–74.
204. 204.
N. Saradha ‘‘Arithmetische Untersuchungen elliptischer Integrale,’’ Math. Ann. 113 (1936), pp. 1–13.Google Scholar
205. 205.
N. Saradha ‘‘Zur Theorie der Abelschen Funktionen und Integrale,’’ J. reine angew. Math. 183 (1941), pp. 110–128.
206. 206.
N. Saradha ‘‘Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise,’’ Math. Ann. 121 (1949), pp. 141–140.
207. 207.
N. Saradha – Einführung in die transzendenten Zahlen, Springer-Verlag, Berlin, 1957.Google Scholar
208. 208.
J.-P. Serre – Cours d’arithmétique, Collection SUP: ‘‘Le Mathématicien,’’ vol. 2, Presses Universitaires de France, Paris, 1970, reprinted 1977. Engl. transl.: A course in arithmetic, Graduate Texts in Mathematics, Vol. 7. Springer-Verlag, New York, 1978.Google Scholar
209. 209.
S. O. Shestakov ‘‘On the measure of algebraic independence of some numbers,’’ Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1992), pp. 8–12, 111.Google Scholar
210. 210.
C. L. Siegel ‘‘Über die Perioden elliptischer Funktionen,’’ J. f. M. 167 (1932), pp. 62–69.Google Scholar
211. 211.
C. L. Siegel – Transcendental Numbers, Annals of Mathematics Studies, no. 16, Princeton University Press, Princeton, N. J., 1949.Google Scholar
212. 212.
J. H. Silverman – The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.Google Scholar
213. 213.
J. H. Silverman – Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994.Google Scholar
214. 214.
A. A. Šmelev ‘‘The algebraic independence of the values of the exponential and an elliptic function,’’ Mat. Zametki 20 (1976), no. 2, pp. 195–202.
215. 215.
A. A. Šmelev ‘‘Algebraic independence of several numbers connected with exponential and elliptic functions,’’ Ukrain. Mat. Zh. 33 (1981), no. 2, pp. 277–282.
216. 216.
M. Takeuchi ‘‘Quantitative results of algebraic independence and Baker’s method,’’ Acta Arith. 119 (2005), no. 3, pp. 211–241.
217. 217.
R. Tijdeman ‘‘On the algebraic independence of certain numbers,’’ Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), pp. 146–162.
218. 218.
M. Toyoda & T. Yasuda ‘‘On the algebraic independence of certain numbers connected with the exponential and the elliptic functions,’’ Tokyo J. Math. 9 (1986), no. 1, pp. 29–40.
219. 219.
R. Tubbs ‘‘A transcendence measure for some special values of elliptic functions,’’ Proc. Amer. Math. Soc. 88 (1983), no. 2, pp. 189–196.
220. 220.
R. Tubbs ‘‘On the measure of algebraic independence of certain values of elliptic functions,’’ J. Number Theory 23 (1986), no. 1, pp. 60–79.
221. 221.
R. Tubbs ‘‘Algebraic groups and small transcendence degree. I,’’ J. Number Theory 25 (1987), no. 3, pp. 279–307.
222. 222.
R. Tubbs ‘‘Elliptic curves in two-dimensional abelian varieties and the algebraic independence of certain numbers,’’ Michigan Math. J. 34 (1987), no. 2, pp. 173–182.
223. 223.
R. Tubbs ‘‘A Diophantine problem on elliptic curves,’’ Trans. Amer. Math. Soc. 309 (1988), no. 1, pp. 325–338.
224. 224.
R. Tubbs ‘‘Algebraic groups and small transcendence degree. II,’’ J. Number Theory 35 (1990), no. 2, pp. 109–127.
225. 225.
K. G. Vasil′ev ‘‘On the algebraic independence of the periods of Abelian integrals,’’ Mat. Zametki 60 (1996), no. 5, pp. 681–691, 799.Google Scholar
226. 226.
I. Wakabayashi ‘‘Algebraic values of meromorphic functions on Riemann surfaces,’’ J. Number Theory 25 (1987), no. 2, pp. 220–229.
227. 227.
I. Wakabayashi – ‘‘Algebraic values of functions on the unit disk,’’ in Prospects of mathematical science (Tokyo, 1986), World Sci. Publishing, Singapore, 1988, pp. 235–266.Google Scholar
228. 228.
I. Wakabayashi – ‘‘An extension of the Schneider-Lang theorem,’’ in Seminar on Diophantine Approximation (Japanese) (Yokohama, 1987), Sem. Math. Sci., vol. 12, Keio Univ., Yokohama, 1988, pp. 79–83.Google Scholar
229. 229.
M. Waldschmidt ‘‘Propriétés arithmétiques des valeurs de fonctions méromorphes algébriquement indépendantes,’’ Acta Arith. 23 (1973), pp. 19–88.
230. 230.
M. Waldschmidt – Nombres transcendants, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 402.
231. 231.
M. Waldschmidt – ‘‘Les travaux de G. V. Chudnovsky sur les nombres transcendants,’’ in Séminaire Bourbaki, Vol. 1975/76, 28e année, Exp. No. 488, Springer, Berlin, 1977, pp. 274–292. Lecture Notes in Math., Vol. 567.Google Scholar
232. 232.
M. Waldschmidt ‘‘Nombres transcendants et fonctions sigma de Weierstrass,’’ C. R. Math. Rep. Acad. Sci. Canada 1 (1978/79), no. 2, pp. 111–114.
233. 233.
M. Waldschmidt – Nombres transcendants et groupes algébriques, Astérisque, vol. 69, Société Mathématique de France, Paris, 1979, With appendices by Daniel Bertrand and Jean-Pierre Serre.Google Scholar
234. 234.
M. Waldschmidt ‘‘Diophantine properties of the periods of the Fermat curve,’’ in Number theory related to Fermat’s last theorem, Proc. Conf., Prog. Math. 26, 79–88 , 1982.Google Scholar
235. 235.
M. Waldschmidt – ‘‘Algebraic independence of transcendental numbers. Gel′fond’s method and its developments,’’ in Perspectives in mathematics, Birkhäuser, Basel, 1984, pp. 551–571.Google Scholar
236. 236.
M. Waldschmidt ‘‘Algebraic independence of values of exponential and elliptic functions,’’ J. Indian Math. Soc. (N.S.) 48 (1984), no. 1–4, pp. 215–228 (1986).
237. 237.
M. Waldschmidt ‘‘Groupes algébriques et grands degrés de transcendance,’’ Acta Math. 156 (1986), no. 3–4, pp. 253–302, With an appendix by J. Fresnel.
238. 238.
M. Waldschmidt – ‘‘Some transcendental aspects of Ramanujan’s work,’’ in Proceedings of the Ramanujan Centennial International Conference (Annamalainagar, 1987), RMS Publ., vol. 1, Ramanujan Math. Soc., 1988, pp. 67–76.Google Scholar
239. 239.
M. Waldschmidt ‘‘Sur la nature arithmétique des valeurs de fonctions modulaires,’’ Astérisque 245 (1997), p. Exp. No. 824, 3, 105–140, Séminaire Bourbaki, Vol. 1996/97.
240. 240.
M. Waldschmidt ‘‘Density measure of rational points on abelian varieties,’’ Nagoya Math. J. 155 (1999), pp. 27–53.
241. 241.
M. Waldschmidt – ‘‘Transcendance et indépendance algébrique de valeurs de fonctions modulaires,’’ in Number theory (Ottawa, ON, 1996), CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI, 1999, pp. 353–375.Google Scholar
242. 242.
M. Waldschmidt – ‘‘Algebraic independence of transcendental numbers: a survey,’’ in Number theory, Trends Math., Birkhäuser and Hindustan Book Agency, Basel and New-Delhi, 2000, pp. 497–527.Google Scholar
243. 243.
M. Waldschmidt ‘‘Transcendence of periods: the state of the art,’’ Pure and Applied Mathematics Quarterly 2 (2006), no. 2, pp. 199–227.
244. 244.
E. Whittaker & G. Watson – A course of modern analysis. An introduction to the general theory on infinite processes and of analytic functions; with an account of the principal transcendental functions. 4th ed., reprinted, Cambridge: At the University Press. 608 p., 1962.Google Scholar
245. 245.
J. Wolfart & G. Wüstholz ‘‘Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen,’’ Math. Ann. 273 (1985), no. 1, pp. 1–15.
246. 246.
G. Wüstholz ‘‘Algebraische Unabhängigkeit von Werten von Funktionen, die gewissen Differentialgleichungen genügen,’’ J. reine angew. Math. 317 (1980), pp. 102–119.
247. 247.
G. Wüstholz ‘‘Sur l’analogue abélien du théorème de Lindemann,’’ C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, pp. 35–37.
248. 248.
G. Wüstholz ‘‘Über das Abelsche Analogon des Lindemannschen Satzes. I,’’ Invent. Math. 72 (1983), no. 3, pp. 363–388.
249. 249.
G. Wüstholz – ‘‘Recent progress in transcendence theory,’’ in Number theory, Noordwijkerhout 1983, Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 280–296.Google Scholar
250. 250.
G. Wüstholz ‘‘Transzendenzeigenschaften von Perioden elliptischer Integrale,’’ J. reine angew. Math. 354 (1984), pp. 164–174.
251. 251.
G. Wüstholz ‘‘Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen,’’ Ann. of Math. (2) 129 (1989), no. 3, pp. 501–517.
252. 252.
A. Y. Yanchenko ‘‘On the measure of algebraic independence of values of derivatives of a modular function (the p-adic case),’’ Mat. Zametki 61 (1997), no. 3, pp. 431–440, Engl. Transl. Math. Notes 61, 3 (1997), 352–359.
253. 253.
K. R. Yu ‘‘Linear forms in elliptic logarithms,’’ J. Number Theory 20 (1985), no. 1, pp. 1–69.
254. 254.
D. Zagier – ‘‘Introduction to modular forms,’’ in From Number Theory to Physics (Les Houches, 1989), Springer, Berlin, 1992, pp. 238–291.Google Scholar