Diophantine approximation on algebraic varieties
Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 439-502.

Nous donnons un aperçu de progrès récents en théorie de l'approximation diophantienne. Le point de départ étant le théorème de Roth, nous nous intéressons d'abord à la conjecture de Mordell, puis ensuite à des résultats analogues en dimension supérieure, résultats dûs à Faltings-Wustholz et à Faltings.

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.

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     title = {Diophantine approximation on algebraic varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {439--502},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {2},
     year = {1999},
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     url = {http://www.numdam.org/item/JTNB_1999__11_2_439_0/}
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Nakamaye, Michael. Diophantine approximation on algebraic varieties. Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 439-502. http://www.numdam.org/item/JTNB_1999__11_2_439_0/

[B1] E. Bombieri, On the Thue-Siegel-Dyson theorem. Acta Math. 148 (1982), 255-296. | MR | Zbl

[B2] E. Bombieri, The Mordell Conjecture revisited. Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV, 17 (1991), 615-640. | Numdam | MR | Zbl

[D] F.J. Dyson, The approximation to algebraic numbers by rationals, Acta Math. 9 (1947), 225-240. | MR | Zbl

[EE] B. Edixhoven & J.-H. Evertse editors, Diophantine Approximation and Abelian Varieties. Springer Lecture Notes 1566 (1993). | MR | Zbl

[EV] H. Esnault & E. Viehweg, Dyson's Lemma for polynomials in several variables (and the Theorem of Roth). Inv. Math. 78 (1984), 445-490. | MR | Zbl

[F1] G. Faltings, Diophantine Approximation on Abelian Varieties. Annals of Math. 133 (1991), 549-576. | MR | Zbl

[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. | MR | Zbl

[FW1] G. Faltings & G. Wüstholz, editors, Rational Points. Vieweg, (1984). | MR | Zbl

[FW2] G. Faltings & G. Wüstholz, Diophantine approximations on projective spaces. Inv. math. 116 (1994), 109-138. | MR | Zbl

[H] M. Hindry, Sur les Conjectures de Mordell et Lang. Astérisque, 209 (1992), 39-56. | Numdam | MR | Zbl

[L1] S. Lang, Fundamentals of Diophantine Geometry. Springer Verlag, (1983). | MR | Zbl

[L2] S. Lang (Ed.), Number Theory III: Diophantine Geometry. Springer Verlag, (1991). | MR | Zbl

[M] D. Mumford, A Remark on Mordell's Conjecture. American Journal of Math. 87, No. 4 (1965), 1007-1016. | MR | Zbl

[N1] M. Nakamaye, Dyson's Lemma and a Theorem of Esnault and Viehweg. Inv. Math. 121 (1995), 355-377. | MR | Zbl

[N2] M. Nakamaye, Dyson's Lemma with Moving Parts. Mathematische Annalen, 310 (1998), 161-168. | MR | Zbl

[N3] M. Nakamaye, Intersection Theory and Diophantine Approximation. to appear, Journal of Algebraic Geometry. | MR | Zbl

[S1] W. Schmidt, Diophantine Approximation, Springer Lecture Notes 785 (1980). | MR | Zbl

[S2] W. Schmidt, Diophantine Approximations and Diophantine Equations. Springer Lecture Notes 1467 (1991). | MR | Zbl

[SE] J.-P. Serre, Lectures on the Mordell-Weil Theorem. Vieweg, (1990). | Zbl

[Vi] C. Viola, On Dyson's lemma. Ann. Sc. Norm. Super. Pisa, 12 (1985), 105-135. | Numdam | MR | Zbl

[V1] P. Vojta, Dyson's lemma for products of two curves of arbitrary genus. Inv. Math. 98 (1989), 107-113. | MR | Zbl

[V2] P. Vojta, Siegel's theorem in the compact case. Annals of Math. 133 (1991), 509-548. | MR | Zbl

[V3] P. Vojta, A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. Journal AMS, 4 (1992), 763-804. | MR | Zbl

[V4] P. Vojta, Some applications of arithmetic algebraic geometry to diophantine approximations. Proceedings of the CIME Conference, nento, (1991), LNM 1553, Springer, ((1993)). | MR | Zbl

[V5] P. Vojta, Integral points on subvarieties of semi-abelian varieties, I. Inv. Math. 126 (1996), 133-181. | MR | Zbl