Integral points on arithmetic surfaces. (Points entiers sur les surfaces arithmétiques.) (French) Zbl 1007.11041

Autissier, Pascal

J. Reine Angew. Math. 531, 201-235 (2001).

Author’s summary: An effective version of R. S. Rumely’s theorem [J. Reine Angew. Math. 368, 127–133 (1986; Zbl 0581.14014)] is proved: a lot of integral points can be found on open sets of arithmetic surfaces with a bound of their heights. For this, S.-W. Zhang’s arithmetic ampleness theorem [Ann. Math. (2) 136, 569–587 (1992; Zbl 0788.14017)] is put into the context of J.-B. Bost’s extension of Arakelov geometry [Ann. Sci. Éc. Norm. Supér. (4) 32, 241–312 (1999; Zbl 0931.14014)]. Moreover, this leads to a generalization of the Fekete-Szegő theorem.

MSC:

11G50	Heights
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

Rumely’s theorem; Zhang’s arithmetic ampleness theorem; Bost’s extension of Arakelov geometry; integral points; arithmetic surfaces; heights; generalization of the Fekete-Szegő theorem