Hostname: page-component-5d59c44645-mrcq8 Total loading time: 0 Render date: 2024-02-22T18:40:21.602Z Has data issue: false hasContentIssue false

The yoga of the Cassels–Tate pairing

Published online by Cambridge University Press:  01 November 2010

Tom Fisher
DPMMS, Centre for Mathematical Sciences, Wilberforce Road Cambridge CB3 0WB, United Kingdom (email:
Edward F. Schaefer
Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, USA (email:
Michael Stoll
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (email:


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Research Article
Copyright © London Mathematical Society 2010


[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) 95112.Google Scholar
[3] Cassels, J. W. S., ‘Second descents for elliptic curves’, J. reine angew. Math. 494 (1998) 101127.Google Scholar
[4] Fisher, T. A., ‘The Cassels–Tate pairing and the Platonic solids’, J. Number Theory 98 (2003) 105155.Google Scholar
[5] Milne, J. S., Arithmetic duality theorems (Academic Press, Boston, MA, 1986).Google Scholar
[6] Poonen, B. and Stoll, M., ‘The Cassels–Tate pairing on polarized abelian varieties’, Ann. of Math. (2) 150 (1999) 11091149.Google Scholar
[7] Schaefer, E. F., ‘2-descent on the Jacobians of hyperelliptic curves’, J. Number Theory 51 (1995) no. 2, 219232.Google Scholar
[8] Schaefer, E. F., ‘Computing a Selmer group of a Jacobian using functions on the curve’, Math. Ann. 310 (1998) no. 3, 447471.Google Scholar
[9] Serre, J.-P., Local fields (Springer, New York, 1979).Google Scholar
[10] Serre, J.-P., ‘Local class field theory’, Algebraic number theory (eds Cassels, J. W. S. and Fröhlich, A.; Academic Press, London, 1967) 129161.Google Scholar
[11] Serre, J.-P., Galois cohomology (Springer, Berlin, 2002).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