Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties

Huayi Chen (1) , Atsushi Moriwaki (2)

1 IF - Institut Fourier
2 Faculty of Science [Kyoto]

In this paper, we study obstructions to the Dirichlet property by two approaches~: density of non-positive points and functionals on adelic $\mathbb R$-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic $\mathbb R$-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.

Mots clés: Arakelov geometry; arithmetic dynamical system; height

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

Huayi Chen, Atsushi Moriwaki. Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties. International Mathematics Research Notices, 2015, 2015 (24), pp.13669-13715. ⟨hal-00950741⟩