  In one of the fundamental results of Arakelov's arithmetic intersection
theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for
arithmetic surfaces by relating the intersection pairing to the negative of the
N\'eron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized
this to higher dimension. In this work, we extend these results to projective
varieties over transcendence degree one function fields. The new challenge is
dealing with non-constant but numerically trivial line bundles coming from the
constant field via Chow's $K/k$-trace functor. As an application of the Hodge
Index Theorem, we also prove a rigidity theorem for the set of canonical height
zero points of polarized algebraic dynamical systems over function fields. For
function fields over finite fields, this gives a rigidity theorem for
preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and
Yuan-Zhang.
