ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 14G05 Peyre, Emmanuel, Université de Strasbourg, France Heights and Tamagawa measures If V is a smooth geometrically integral variety over a number field k such that the anticanonical sheaf ω_{V}^{−1} is very ample, then any basis of the space of sections Γ(V, ω_{V}^{−1}) defines an embedding of V in a projective space and a height on V (k). Manin had conjectured that, in general, the asymptotic behavior of the number of points of bounded height on V should have the form

CH(log H)^{t−1} when H → +∞

where t is the rank of the Picard group of V and C a real constant. An empirical formula for C will be presented with various evidences in its favor. Reference: E. Peyre, Hauteurs et mesures de Tamgawa sur les variétés de Fano, Duke Math. J. 79 (1995) 101–218