Igusa integrals and volume asymptotics in analytic and adelic geometry

Antoine Chambert-Loir (1) , Yuri Tschinkel

1 IRMAR - Institut de Recherche Mathématique de Rennes

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.

Mots clés: Heights; Poisson formula; Manin's conjecture; Tamagawa measure

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Antoine Chambert-Loir, Yuri Tschinkel. Igusa integrals and volume asymptotics in analytic and adelic geometry. Confluentes Mathematici, 2010, 2 (3), pp.351-429. ⟨10.1142/S1793744210000223⟩. ⟨hal-00455582⟩