Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2024 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Old and new conjectured diophantine inequalities
HTML articles powered by AMS MathViewer

by Serge Lang PDF
Bull. Amer. Math. Soc. 23 (1990), 37-75
References
  • 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 $x^{3}-y^{2}$, Norske Vid. Selsk. Forh. (Trondheim) 38 (1965), 65–69. MR 186620
  • P. E. Blanksby and H. L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971), 355–369. MR 296021, DOI 10.4064/aa-18-1-355-369
    • [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 $x^{3}-y^{2}=k$ and a conjecture of M. Hall, Mat. Zametki 32 (1982), no. 3, 273–275, 425 (Russian). MR 677595
  • H. Davenport, On $f^{3}\,(t)-g^{2}\,(t)$, Norske Vid. Selsk. Forh. (Trondheim) 38 (1965), 86–87. MR 186621
  • 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, DOI 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 434947, DOI 10.1112/plms/s3-33.2.193
  • 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, DOI 10.1007/978-1-4612-4752-4
  • 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, DOI 10.1007/978-3-662-07010-9
  • Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
  • 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, DOI 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, DOI 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, DOI 10.1017/CBO9780511752490
  • R. C. Mason, The hyperelliptic equation over function fields, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 2, 219–230. MR 691990, DOI 10.1017/S0305004100060497
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
  • André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI 10.1007/bf02684271
  • K. A. Ribet, On modular representations of $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$ arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476. MR 1047143, DOI 10.1007/BF01231195
  • Jean-Pierre Serre, Sur les représentations modulaires de degré $2$ de $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Duke Math. J. 54 (1987), no. 1, 179–230 (French). MR 885783, DOI 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 419359, DOI 10.1007/BF01389745
  • 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) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
  • Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Similar Articles
Additional Information
  • Journal: Bull. Amer. Math. Soc. 23 (1990), 37-75
  • MSC (1985): Primary 11D41, 11D75; Secondary 11G05, 11G30
  • DOI: https://doi.org/10.1090/S0273-0979-1990-15899-9
  • MathSciNet review: 1005184