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

  • Timo KellerEmail author


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 \(E_8\).


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

Mathematics Subject Classification

11S25 14F20 14L17 



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


  1. 1.
    Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21, x, 325 p. Springer, Berlin (1990)CrossRefGoogle Scholar
  2. 2.
    Colliot-Thélène, J.-L., Parimala, R., Suresh, V.: Patching and local-global principles for homogeneous spaces over function fields of \(p\)-adic curves. Comment. Math. Helv. 87(4), 1011–1033 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fu, L.: Étale cohomology theory. Nankai Tracts in Mathematics, vol. 13, ix 611 p. Hackensack: World Scientific (2011)Google Scholar
  4. 4.
    Görtz, U., Wedhorn, T.: Algebraic geometry I. Schemes. With examples and exercises. Advanced Lectures in Mathematics, vii, 615 p. Vieweg+Teubner, Wiesbaden (2010)CrossRefGoogle Scholar
  5. 5.
    Harari, D., Szamuely, T.: Arithmetic duality theorems for 1-motives. J. Reine Angew. Math. 578, 93–128 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Harari, D., Szamuely, T.: Local-global questions for tori over \(p\)-adic function fields. J. Algebr. Geom. 25(3), 571–605 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Izquierdo, D.: Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux. Doc. Math. 22, 297–361 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Liu, Q.: Algebraic geometry and arithmetic curves. Translated by Erné, R. Oxford Graduate Texts in Mathematics, vol. 6, xv, 577 p. Oxford University Press, Oxford (2006)Google Scholar
  9. 9.
    Milne, J.S.: Étale cohomology. Princeton Mathematical Series, vol. 33, XIII, 323 p. Princeton University Press, Princeton (1980)Google Scholar
  10. 10.
    Milne, J.S.: Arithmetic duality theorems. Perspectives in Mathematics, vol. 1, 2nd edn, viii, 339 p. BookSurge, LLC, Charleston, SC (2006)Google Scholar
  11. 11.
    Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, 2nd edn. Springer, Berlin. (2008)
  12. 12.
    Serre, J.-P.: Local fields. Graduate Texts in Mathematics, vol. 67. Springer, Berlin (1979)CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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