Arakelov geometry offers a framework to develop an arithmetic counterpart of the usual intersection theory. In fact, for varieties defined over the ring of integers of a number field, and inspired by the geometric case, one can define a suitable notion of arithmetic Chow groups and of an arithmetic intersection product. In this talk, I will begin with a brief introduction to Arakelov geometry in the flavour of Gillet-Soulé, and then present a joint work with R. Gualdi (Universitat Politècnica de Catalunya). We prove an arithmetic analogue of the classical Shioda-Tate formula, relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety to some of its geometric invariants. In doing so, we also characterize numerically trivial arithmetic divisors, partially confirming a conjecture by Gillet and Soulé.