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 \Lambda 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 \Lambda L) is not trivial in Pic(Z)? 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
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