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

  • Timo Keller
Article
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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 Open image in new window for F a finite étale 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 \(E_8\).

Keywords

Galois cohomology Étale and other Grothendieck topologies and cohomologies Affine algebraic groups 

Mathematics Subject Classification

11S25 14F20 14L17 

Notes

Acknowledgements

I thank Diego Izquierdo for suggesting this problem to me, and Ulrich Görtz and Diego Izquierdo for helpful discussions.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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