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The yoga of the Cassels–Tate pairing
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Arithmetic algebraic geometry
Published online by Cambridge University Press: 01 November 2010
Abstract
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Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels–Tate pairing. In this article, we prove that the two pairings are the same.
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- Research Article
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- Copyright © London Mathematical Society 2010
References
[1] Brown, K. S., Cohomology of groups, Graduate Texts in Mathematics 87 (Springer, New York, 1994).Google Scholar
[2] Cassels, J. W. S., ‘Arithmetic on curves of genus 1, IV. Proof of the Hauptvermutung’, J. reine angew. Math. 211 (1962) 95–112.CrossRefGoogle Scholar
[3] Cassels, J. W. S., ‘Second descents for elliptic curves’, J. reine angew. Math. 494 (1998) 101–127.CrossRefGoogle Scholar
[4] Fisher, T. A., ‘The Cassels–Tate pairing and the Platonic solids’, J. Number Theory 98 (2003) 105–155.CrossRefGoogle Scholar
[6] Poonen, B. and Stoll, M., ‘The Cassels–Tate pairing on polarized abelian varieties’, Ann. of Math. (2) 150 (1999) 1109–1149.CrossRefGoogle Scholar
[7] Schaefer, E. F., ‘2-descent on the Jacobians of hyperelliptic curves’, J. Number Theory 51 (1995) no. 2, 219–232.CrossRefGoogle Scholar
[8] Schaefer, E. F., ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998) no. 3, 447–471.CrossRefGoogle Scholar
[10] Serre, J.-P., ‘Local class field theory’, Algebraic number theory (eds Cassels, J. W. S. and Fröhlich, A.; Academic Press, London, 1967) 129–161.Google Scholar
[12] Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer, New York, 1992).Google Scholar
[13] Stamminger, S., ‘Explicit 8-descent on elliptic curves’, PhD Thesis, International University Bremen, 2005.Google Scholar
[14] Swinnerton-Dyer, H. P. F., ‘ 2n-descent on elliptic curves for all n’, J. London Math. Soc. (2), to appear.Google Scholar
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