  Let $X$ be an arithmetic variety over the ring of integers of a number field
$K$, with smooth generic fiber $X_K$. We give a formula that relates the
dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil
rank of the Albanese variety of $X_K$ and the rank of the N\'eron-Severi group
of $X_K$. This is a higher dimensional and arithmetic version of the classical
Shioda-Tate formula for elliptic surfaces. Such analogy is strengthened by the
fact that we show that the numerically trivial arithmetic $\mathbb{R}$-divisors
on $X$ are exactly the linear combinations of principal ones. This result is
equivalent to the non-degeneracy of the arithmetic intersection pairing in the
argument of divisors, partially confirming [GS94, Conjecture 1].
