Skip to main content

Applications of arithmetic algebraic geometry to diophantine approximations

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1553)

Keywords

  • Line Sheaf
  • Abelian Variety
  • Global Section
  • Diophantine Approximation
  • Generic Fiber

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. S. Abhyankar, Resolution of singularities of arithmetical surfaces, Arithmetical algebraic geometry (O. F. G. Schilling, ed.), Harper & Row, New York, 1965, pp. 111–152.

    Google Scholar 

  2. M. Artin, Lipman's proof of resolution of singularities for surfaces, Arithmetic geometry (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, New York, 1986, pp. 267–287.

    CrossRef  Google Scholar 

  3. V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, AMS Surveys and Monographs 33, Amer. Math. Soc., Providence, R. I., 1990.

    MATH  Google Scholar 

  4. J.-B. Bost, H. Gillet, and C. Soulé, Un analogue arithmétique du théorème de Bezout, C. R. Acad. Sci, Paris, Sér. I 312 (1991), 845–848.

    MATH  Google Scholar 

  5. E. Bombieri, The Mordell conjecture revisited, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV 17 (1990), 615–640.

    MathSciNet  MATH  Google Scholar 

  6. E. Bombieri and J. Vaaler, On Siegel's lemma, Invent. Math. 73 (1983), 11–32; addendum, Invent. Math. 75 (1984), 377.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. F. J. Dyson, The approximation to algebraic numbers by rationals, Acta Math. 79 (1947), 225–240.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. G. Faltings, Diophantine approximation on abelian varieties, Ann. Math. 133 (1991), 549–576.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. _____, The general case of S. Lang's conjecture (to appear).

    Google Scholar 

  10. W. Fulton and S. Lang, Riemann-Roch algebra, Grundlehren der mathematischen Wissenschaften 277, Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  11. M. J. Greenberg, Lectures on forms in many variables, Mathematics lecture notes series, W. A. Benjamin, Inc., New York, 1969.

    Google Scholar 

  12. H. Gillet and C. Soulé, Arithmetic intersection theory, Publ. Math. IHES 72 (1990), 93–174.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. _____, Characteristic classes for algebraic vector bundles with hermitian metric. I, Ann. Math. 131 (1990), 163–203; II, Ann. Math. 131 (1990), 205–238.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. _____, Analytic torsion and the arithmetic Todd genus, Topology 30 (1991), 21–54.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. , Un théorème de Riemann-Roch-Grothendieck arithmétique, C. R. Acad. Sci. Paris, Sér. I 309 (1989), 929–932.

    MATH  Google Scholar 

  16. R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics 156, Springer-Verlag, New York, 1970.

    MATH  Google Scholar 

  17. _____, Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.

    CrossRef  MATH  Google Scholar 

  18. Y. Kawamata, On Bloch's conjecture, Invent. Math. 57 (1980), 97–100.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics 58, Springer-Verlag, New York, 1977.

    CrossRef  MATH  Google Scholar 

  20. L. Lafforgue, Une version en géométrie diophantienne du “lemma de l'indice” (to appear).

    Google Scholar 

  21. S. Lang, Some theorems and conjectures in diophantine equations, Bull. AMS 66 (1960), 240–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. _____, Integral points on curves, Publ. Math. IHES 6 (1960), 27–43.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. _____, Introduction to diophantine approximations, Addison-Wesley, Reading, Mass., 1966.

    MATH  Google Scholar 

  24. _____, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970: reprinted, Springer-Verlag, Berlin-Heidelberg-New York, 1986.

    MATH  Google Scholar 

  25. _____, Fundamentals of diophantine geometry, Springer-Verlag, New York, 1983.

    CrossRef  MATH  Google Scholar 

  26. _____, Hyperbolic and diophantine analysis, Bull. AMS 14 (1986), 159–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. _____, Introduction to Arkelov theory, Springer-Verlag, New York, 1988.

    CrossRef  Google Scholar 

  28. W. J. LeVeque, Topics in Number Theory, Vol. II, Addison-Wesley, Reading, Mass., 1956.

    MATH  Google Scholar 

  29. D. Mumford, A remark on Mordell's conjecture, Amer. J. Math. 87 (1965), 1007–1016.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. J. Noguchi, A higher dimensional analogue of Mordell's conjecture over function fields, Math. Ann. 258 (1981), 207–212.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207–223.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; corrigendum, Mathematika 2 (1955), 168.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. W. M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics 785, Springer-Verlag, Berlin Heidelberg, 1980.

    MATH  Google Scholar 

  34. J. H. Silverman, The theory of height functions, Arithmetic geometry (G. Cornell and J. H. Silverman, eds.), Springer-Verlag, New York, 1986, pp. 151–166.

    CrossRef  Google Scholar 

  35. C. Soulé, Géométrie d'Arakelov et théorie des nombres transcendants, Journées arithmétiques, Luminy, Astérisque (to appear).

    Google Scholar 

  36. C. Soulé, D. Abramovich, J.-F. Burnol, and J. Kramer, Lectures on Arakelov Geometry, Cambridge studies in applied mathematics 33, Cambridge University Press, Cambridge, 1992.

    CrossRef  MATH  Google Scholar 

  37. J. Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257–289.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics 439, Springer-Verlag, Berlin Heidelberg, 1975.

    MATH  Google Scholar 

  39. P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics 1239, Springer-Verlag, Berlin Heidelberg, 1987.

    MATH  Google Scholar 

  40. _____, Dyson's lemma for a product of two curves of arbitrary genus, Invent. Math. 98 (1989), 107–113.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. _____, Mordell's conjecture over function fields, Invent. Math. 98 (1989), 115–138.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. _____, Siegel's theorem in the compact case, Ann. Math. 133 (1991), 509–548.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. _____, A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing, J. Amer. Math. Soc. (to appear).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1993 Springer-Verlag

About this chapter

Cite this chapter

Vojta, P. (1993). Applications of arithmetic algebraic geometry to diophantine approximations. In: Ballico, E. (eds) Arithmetic Algebraic Geometry. Lecture Notes in Mathematics, vol 1553. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084730

Download citation

  • DOI: https://doi.org/10.1007/BFb0084730

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57110-0

  • Online ISBN: 978-3-540-47909-3

  • eBook Packages: Springer Book Archive