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 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 \(E_8\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21, x, 325 p. Springer, Berlin (1990)
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)
Fu, L.: Étale cohomology theory. Nankai Tracts in Mathematics, vol. 13, ix 611 p. Hackensack: World Scientific (2011)
Görtz, U., Wedhorn, T.: Algebraic geometry I. Schemes. With examples and exercises. Advanced Lectures in Mathematics, vii, 615 p. Vieweg+Teubner, Wiesbaden (2010)
Harari, D., Szamuely, T.: Arithmetic duality theorems for 1-motives. J. Reine Angew. Math. 578, 93–128 (2005)
Harari, D., Szamuely, T.: Local-global questions for tori over \(p\)-adic function fields. J. Algebr. Geom. 25(3), 571–605 (2016)
Izquierdo, D.: Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux. Doc. Math. 22, 297–361 (2017)
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)
Milne, J.S.: Étale cohomology. Princeton Mathematical Series, vol. 33, XIII, 323 p. Princeton University Press, Princeton (1980)
Milne, J.S.: Arithmetic duality theorems. Perspectives in Mathematics, vol. 1, 2nd edn, viii, 339 p. BookSurge, LLC, Charleston, SC (2006)
Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, 2nd edn. Springer, Berlin. https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/NSW2.2.pdf (2008)
Serre, J.-P.: Local fields. Graduate Texts in Mathematics, vol. 67. Springer, Berlin (1979)
Acknowledgements
I thank Diego Izquierdo for suggesting this problem to me, and Ulrich Görtz and Diego Izquierdo for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ulf Kühn.
Rights and permissions
About this article
Cite this article
Keller, T. A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields. Abh. Math. Semin. Univ. Hambg. 88, 289–295 (2018). https://doi.org/10.1007/s12188-018-0196-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-018-0196-7