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.
Tyler Lawson’s work is partially supported by NSF DMS-1206008.
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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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Lawson, T., Wuthrich, C. (2016). Vanishing of Some Galois Cohomology Groups for Elliptic Curves. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_11
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DOI: https://doi.org/10.1007/978-3-319-45032-2_11
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