Abstract

Under a certain finiteness assumption of Tate-Shafarevich groups, Kato and Trihan showed the BSD conjecture for abelian varieties over global function fields of positive characteristic. We explain how to obtain the equivariant generalisation of the work of Kato and Trihan for semi-stable abelian varieties with respect to (not necessarily abelian) Galois extensions L/K of global function fields that are at most tamely ramified everywhere. This is a joint work in progress with David Burns and Mahesh Kakde.