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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)

Abstract

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.

Keywords

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

Notes

Acknowledgements

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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