Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber
X
K
{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
{X_{K}}
and the rank of the Néron–Severi group of
X
K
{X_{K}}
.
This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces.
Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic
ℝ
{\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 a conjecture by H. Gillet and C. Soulé.