Line Bundles, Rational Points and Ideal Classes

In this note, we will use the term “arithmetic variety” for a normal scheme X for which the structure morphism f : X → Spec(Z) is proper and flat. Let V be a proper, normal (not necessarily geometrically connected) variety over Q. Let us choose a normal model for V over Z, that is an arithmetic variety X whose generic fiber is identified with V . Suppose that F is a number field and consider the F -rational points of V . These correspond bijectively to R-valued points of X, with R the ring of integers of F . If P is an F -rational point of V , we will also denote by P : Spec(R) → X the corresponding R-valued point of X. Suppose that L is a line bundle on the arithmetic variety X. We say that L is trivial, when it is isomorphic to the structure sheaf OX . We will denote by P ∗L the pull-back of L to Spec(R) via the morphism P ; then P ∗L is a line bundle on Spec(R). It gives an element (P ∗L) in the class group Pic(R) of R. In what follows, we will identify Pic(R) with the ideal class group Cl(F ). This paper is motivated by the following question of the second named author: