Let be a field and let be a fixed algebraic closure of . Let be an elliptic surface over a curve and let be the function field of . Let (or more precisely ). The Néron-Severi group of , denoted by , is by definition the group of divisors on modulo algebraic equivalence. Under the previous assumptions, is a finitely generated abelian group (this is a consequence of the so-called ‘theorem of the base’ which can be found in ). The Néron-Severi group of , denoted by , is simply the image of the group of divisors on in . Let be the subgroup generated by the image of the zero-section and all the irreducible components of the fibers of . is sometimes called the “trivial part” of .
Theorem (Shioda-Tate formula).
For each let be the number of irreducible components on the fiber at , i.e. . Then:
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