Line bundles, rational points and ideal classes

Abstract

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 .I fP 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 )o fR. 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: Question. Suppose that the line bundle L on X is not trivial. Is there a number field F and an F -rational point P of V such that the ideal class (P ∗ L) is not trivial? As a variant of this question, we could also ask: Is there a scheme Z which is finite and flat over Spec(Z) and a morphism P : Z → X such that (P ∗ L )i s not trivial in Pic(Z)? L. Szpiro has informed us that he independently raised this question earlier. If the answer to the question is always positive, then line bundles on arithmetic varieties are characterized by their restrictions to integral points. Here are some interesting facts about this question: 1. The answer is positive when X = P n . Indeed, restricting along any linear morphism P 1 → P n gives an isomorphism on Picard groups. Therefore, it is enough to show the statement for X = P 1 . Take L = OX(n), n � 0. There is a number field F with an ideal A whose ideal class is not n-torsion. We can always write A = aOF + bOF with a, b in OF . Consider the F -rational point (a; b )o fP 1 . This gives a morphism P :S pec(OF ) → P 1F → P 1 .