  Let $\omega$ be a relative differential on aithmetic surface $X$. We
construct a family of rational functions $G_x$ on $X\otimes\Bbb{C}$, which can
approximate local antiderivatives of $\omega$ over an open set on
$X\otimes\Bbb{C}$. From this family of functions, we construct a rational
function $G_2$ on $X$. The function $G_2$ can generate an element in the ring
of integers of a number field, which can approximate an inner product produced
by $\omega$ and the conjugate of $\omega$ over an open set on
$X\otimes\Bbb{C}$. This will give a relation between the height of a rational
curve $E_P$ on $X$ and the canonical norm of $\omega$ on $X\otimes\Bbb{C}$.
This relation will give an upper bound for the height of $E_P$ under a few
assumptions.
