Toward Dirichlet's unit theorem on arithmetic varieties

TL;DR

This paper explores a higher-dimensional analogue of Dirichlet's unit theorem for arithmetic varieties, proposing a fundamental question and providing a partial answer using the arithmetic Hodge index theorem.

Contribution

It introduces a new fundamental question in higher-dimensional number theory and offers a partial solution leveraging the arithmetic Hodge index theorem.

Findings

Proposes a higher-dimensional analogue of Dirichlet's unit theorem.

Provides a partial answer to the fundamental question.

Utilizes the arithmetic Hodge index theorem in the analysis.

Abstract

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.