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Let  X  be an arithmetic variety over the ring of integers of a number field  K , with smooth generic fiberX  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 ofX  K{X_{K}}    and the rank of the Néron–Severi group ofX  K{X_{K}}   . This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces.

Numerical equivalence of ℝ-divisors and Shioda–Tate formula for arithmetic varieties