A decision procedure for certain abelian varieties over function fields
Let \(\mathbb{Q}_ c\) be the algebraic closure of the rational numbers. The authors prove the existence of a decision procedure for the additive group of isogenies between two abelian varieties over \(\mathbb{Q}_ c (t)\). Using Faltings' isogeny theorem, a countable listing of possible isogenies for a sequence of lower bounds, and a calculation of Galois actions on the division points modulo \(m\) for increasing \(m\) for an upper bound, one gets a decision procedure for abelian varieties which become trivial over some finite extension fields. A formula of Kodaira and Shioda for the rank of general elliptic surfaces is proven to be valid for an abelian varieties over \(\mathbb{Q}_ c (t)\) having no continuous family of sections, namely the rank of numerical equivalence classes of 2-dimensional divisors modulo classes from fibres and the zero section. The paper also gives a computable upper bound on the rank based on étale cohomology which is equal to the rank if the Tate conjecture holds.
scientific article; zbMATH DE number 554215