Arithmetic of abelian varieties over function fields and an application to anabelian geometry

We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which we prove are finitely generated abelian groups. We also discuss an application of our investigation to the Grothendieck anabelian section conjecture (joint work with Akio Tamagawa).