Vanishing of Some Galois Cohomology Groups for Elliptic Curves

  • Tyler Lawson
  • Christian Wuthrich
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 188)


Let \(E/\mathbb {Q}\) be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of \(\mathbb {Q}\) obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group \(H^1\bigl ( G, E[p]\bigr )\) does not vanish, and investigate the analogous question for \(E[p^i]\) when \(i>1\). We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald–Wang problem for elliptic curves.


Elliptic curves Galois cohomology Grunwald-Wang problem Birch and Swinnerton-Dyer conjecture 



It is our pleasure to thank Jean Gillibert and John Coates for interesting comments and suggestions. We are also grateful to Brendan Creutz for pointing us to [7].


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Vincent Hall 323, Department of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.School of Mathematical SciencesUniversity of Nottingham, University ParkNottinghamUK

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