A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields

Abstract

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups for F a finite etale Galois module on K of order invertible in K and with $$F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))$$ . Furthermore, we prove that $$\mathrm {H}^1(K,G) = 0$$ for G a simply connected, quasisplit semisimple group over K not of type $$E_8$$ .