Zariski decompositions on arithmetic surfaces

In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic varieties.Comment: 81 pages. Rewrite several part

Toward Dirichlet's unit theorem on arithmetic varieties

In this paper, we would like to propose a fundamental question about a higher dimensional analogue of Dirichlet's unit theorem. We also give a partial answer to the question as an application of the arithmetic Hodge index theorem.Comment: 47 pages, rewrite Subsection 1.2 and Subsection 2.1 for the forthcoming pape