On the arithmetic of abelian varieties

Abstract

Abstract We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated ℤ {\mathbb{Z}} -modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.