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Line bundles, rational points and ideal classes A. Agboola and G. Pappas

Summary: Line bundles, rational points and ideal classes
A. Agboola and G. Pappas
July 18, 2000
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, irreducible (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 R­valued 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
R­valued 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 \Lambda L the pull­back
of L to Spec(R) via the morphism P ; then P \Lambda L is a line bundle on Spec(R). It gives an
element (P \Lambda 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:
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


Source: Agboola, Adebisi - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics