Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Old and new conjectured diophantine inequalities


Author: Serge Lang
Journal: Bull. Amer. Math. Soc. 23 (1990), 37-75
MSC (1985): Primary 11D41, 11D75; Secondary 11G05, 11G30
MathSciNet review: 1005184
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • M. Artin, Néron models, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 213–230. MR 861977
  • B. J. Birch, S. Chowla, Marshall Hall Jr., and A. Schinzel, On the difference 𝑥³-𝑦², Norske Vid. Selsk. Forh. (Trondheim) 38 (1965), 65–69. MR 0186620
  • P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369. MR 0296021
  • [BLSTW] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorization of b ±1, b = 2, 3, 5, 6, 7, 10, 11 up to high powers, Contemporary Mathematics Vol. 22, AMS, 1983.
  • L. V. Danilov, The Diophantine equation 𝑥³-𝑦²=𝑘 and a conjecture of M. Hall, Mat. Zametki 32 (1982), no. 3, 273–275, 425 (Russian). MR 677595
  • H. Davenport, On 𝑓³(𝑡)-𝑔²(𝑡), Norske Vid. Selsk. Forh. (Trondheim) 38 (1965), 86–87. MR 0186621
  • Gerhard Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1, iv+40. MR 853387
  • [Fr2] G. Frey, Links between elliptic curves and solutions of A — B = C, J. Indian Math. Soc., 51 (1987), pp. 117-145.
  • [Ha] M. Hall, The diophantine equation x 3—y2 = k, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, London, 1971, pp. 173-198.
  • M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), no. 2, 419–450. MR 948108, 10.1007/BF01394340
  • Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. MR 0434947
  • Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Compositio Math. 38 (1979), no. 1, 121–128. MR 523268
  • Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960
  • Serge Lang, Conjectured Diophantine estimates on elliptic curves, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 155–171. MR 717593
  • Serge Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin-New York, 1978. MR 518817
  • Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124
  • H. W. Lenstra Jr. and F. Oort, Abelian varieties having purely additive reduction, J. Pure Appl. Algebra 36 (1985), no. 3, 281–298. MR 790619, 10.1016/0022-4049(85)90079-9
  • R. C. Mason, Equations over function fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 149–157. MR 756091, 10.1007/BFb0099449
  • R. C. Mason, Diophantine equations over function fields, London Mathematical Society Lecture Note Series, vol. 96, Cambridge University Press, Cambridge, 1984. MR 754559
  • R. C. Mason, The hyperelliptic equation over function fields, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 2, 219–230. MR 691990, 10.1017/S0305004100060497
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). MR 488287
  • André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ.Math. No. 21 (1964), 128 (French). MR 0179172
  • K. A. Ribet, On modular representations of 𝐺𝑎𝑙(\overline{𝑄}/𝑄) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, 10.1007/BF01231195
  • Jean-Pierre Serre, Sur les représentations modulaires de degré 2 de 𝐺𝑎𝑙(\overline{𝑄}/𝑄), Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, 10.1215/S0012-7094-87-05413-5
  • Joseph H. Silverman, Lower bound for the canonical height on elliptic curves, Duke Math. J. 48 (1981), no. 3, 633–648. MR 630588
  • John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 0419359
  • J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33–52. Lecture Notes in Math., Vol. 476. MR 0393039
  • Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 11D41, 11D75, 11G05, 11G30

Retrieve articles in all journals with MSC (1985): 11D41, 11D75, 11G05, 11G30


Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1990-15899-9