Diophantine approximation on algebraic varieties

Michael Nakamaye

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 2, page 439-502
  • ISSN: 1246-7405

Abstract

top
We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.

How to cite

top

Nakamaye, Michael. "Diophantine approximation on algebraic varieties." Journal de théorie des nombres de Bordeaux 11.2 (1999): 439-502. <http://eudml.org/doc/248322>.

@article{Nakamaye1999,
abstract = {We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.},
author = {Nakamaye, Michael},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Diophantine approximation; rational points on algebraic varieties; arithmetic algebraic geometry; Roth's theorem; nonvanishing lemma for polynomials in several variables; Roth's lemma; Dyson's lemma; Mordell conjecture; Faltings' theorem; finiteness of rational points; algebraic curve of genus greater than one; Vojta's generalization of Dyson's lemma; products of curves of arbitrary genus; Lang conjecture; Subspace Theorem; lower bound for the rational approximation to a hyperplane},
language = {eng},
number = {2},
pages = {439-502},
publisher = {Université Bordeaux I},
title = {Diophantine approximation on algebraic varieties},
url = {http://eudml.org/doc/248322},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Nakamaye, Michael
TI - Diophantine approximation on algebraic varieties
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 2
SP - 439
EP - 502
AB - We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.
LA - eng
KW - Diophantine approximation; rational points on algebraic varieties; arithmetic algebraic geometry; Roth's theorem; nonvanishing lemma for polynomials in several variables; Roth's lemma; Dyson's lemma; Mordell conjecture; Faltings' theorem; finiteness of rational points; algebraic curve of genus greater than one; Vojta's generalization of Dyson's lemma; products of curves of arbitrary genus; Lang conjecture; Subspace Theorem; lower bound for the rational approximation to a hyperplane
UR - http://eudml.org/doc/248322
ER -

References

top
  1. [B1] E. Bombieri, On the Thue-Siegel-Dyson theorem. Acta Math.148 (1982), 255-296. Zbl0505.10015MR666113
  2. [B2] E. Bombieri, The Mordell Conjecture revisited. Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV, 17 (1991), 615-640. Zbl0722.14010MR1093712
  3. [D] F.J. Dyson, The approximation to algebraic numbers by rationals, Acta Math.9 (1947), 225-240. Zbl0030.02101MR23854
  4. [EE] B. Edixhoven & J.-H. Evertse editors, Diophantine Approximation and Abelian Varieties. Springer Lecture Notes 1566 (1993). MR1288998
  5. [EV] H. Esnault & E. Viehweg, Dyson's Lemma for polynomials in several variables (and the Theorem of Roth). Inv. Math.78 (1984), 445-490. Zbl0545.10021MR768988
  6. [F1] G. Faltings, Diophantine Approximation on Abelian Varieties. Annals of Math.133 (1991), 549-576. Zbl0734.14007MR1109353
  7. [F2] G. Faltings, The general case of S. Lang's conjecture. in: Christante and Messing (eds.), Barsotti symposium in algebraic geometry, Academic Press, (1994), 175-182. Zbl0823.14009MR1307396
  8. [FW1] G. Faltings & G. Wüstholz, editors, Rational Points. Vieweg, (1984). Zbl0753.14019MR766568
  9. [FW2] G. Faltings & G. Wüstholz, Diophantine approximations on projective spaces. Inv. math.116 (1994), 109-138. Zbl0805.14011MR1253191
  10. [H] M. Hindry, Sur les Conjectures de Mordell et Lang. Astérisque, 209 (1992), 39-56. Zbl0792.14009MR1211002
  11. [L1] S. Lang, Fundamentals of Diophantine Geometry. Springer Verlag, (1983). Zbl0528.14013MR715605
  12. [L2] S. Lang (Ed.), Number Theory III: Diophantine Geometry. Springer Verlag, (1991). Zbl0744.14012MR1112552
  13. [M] D. Mumford, A Remark on Mordell's Conjecture. American Journal of Math.87, No. 4 (1965), 1007-1016. Zbl0151.27301MR186624
  14. [N1] M. Nakamaye, Dyson's Lemma and a Theorem of Esnault and Viehweg. Inv. Math.121 (1995), 355-377. Zbl0855.11036MR1346211
  15. [N2] M. Nakamaye, Dyson's Lemma with Moving Parts. Mathematische Annalen, 310 (1998), 161-168. Zbl0940.11026MR1600039
  16. [N3] M. Nakamaye, Intersection Theory and Diophantine Approximation. to appear, Journal of Algebraic Geometry. Zbl0953.11026MR1658224
  17. [S1] W. Schmidt, Diophantine Approximation, Springer Lecture Notes 785 (1980). Zbl0421.10019MR568710
  18. [S2] W. Schmidt, Diophantine Approximations and Diophantine Equations. Springer Lecture Notes 1467 (1991). Zbl0754.11020MR1176315
  19. [SE] J.-P. Serre, Lectures on the Mordell-Weil Theorem. Vieweg, (1990). Zbl0676.14005
  20. [Vi] C. Viola, On Dyson's lemma. Ann. Sc. Norm. Super. Pisa, 12 (1985), 105-135. Zbl0596.10032MR818804
  21. [V1] P. Vojta, Dyson's lemma for products of two curves of arbitrary genus. Inv. Math.98 (1989), 107-113. Zbl0666.10024MR1010157
  22. [V2] P. Vojta, Siegel's theorem in the compact case. Annals of Math.133 (1991), 509-548. Zbl0774.14019MR1109352
  23. [V3] P. Vojta, A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. Journal AMS, 4 (1992), 763-804. Zbl0778.11037MR1151542
  24. [V4] P. Vojta, Some applications of arithmetic algebraic geometry to diophantine approximations. Proceedings of the CIME Conference, nento, (1991), LNM 1553, Springer, ((1993)). Zbl0846.14009MR1338861
  25. [V5] P. Vojta, Integral points on subvarieties of semi-abelian varieties, I. Inv. Math.126 (1996), 133-181. Zbl1011.11040MR1408559

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.