A Decision Procedure for Certain Abelian Varieties over Function Fields

AbstractWe prove that there is a decision procedure for the additive group of isogenies between two abelian varieties over Qc(t), where Qc is the algebraic closure of the rational numbers. This procedure uses Falting′s 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. This gives a decision procedure for abelian varieties which become trivial over some finite extension field. We prove that a formula of Kodaira and Shioda for rank of general elliptic surfaces is valid for all abelian varieties over Qc(t) having no continuous family of sections, namely rank of numerical equivalence classes of 2-dimensional divisors modulo classes from fibres and the zero section. We also give a computable upper bound on the rank based on étale cohomology. It is equal to the rank if the Tate conjecture holds