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

Abstract

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.