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

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 Neron-Seven group of 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-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. Soule.

Automated summary

The paper presents a formula relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety X with the Mordell-Weil rank of the Albanese variety of X-K and the rank of the Neron-Seven group of X-K. Additionally, it proves that the numerically trivial arithmetic R-divisors on X are exactly the linear combinations of principal ones.