  This is the first paper of a series. We prove an arithmetic Hodge index
theorem for adelic line bundles on projective varieties over number fields. It
extends the arithmetic Hodge index theorem of Faltings, Hriljac and Moriwaki on
arithmetic varieties. As consequences, we obtain the uniqueness part of the
non-archimedean Calabi--Yau theorem, and a rigidity property of the sets of
preperiodic points of polarizable algebraic dynamical systems over number
fields.
