Hostname: page-component-f554764f5-c4bhq Total loading time: 0 Render date: 2025-04-12T12:07:24.228Z Has data issue: false hasContentIssue false
  • English
  • Français

The yoga of the Cassels–Tate pairing

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 November 2010

Tom Fisher ,
Edward F. Schaefer  and
Michael Stoll
Tom Fisher
Affiliation:
DPMMS, Centre for Mathematical Sciences, Wilberforce Road Cambridge CB3 0WB, United Kingdom (email: T.A.Fisher@dpmms.cam.ac.uk)
Edward F. Schaefer
Affiliation:
Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, USA (email: eschaefer@scu.edu)
Michael Stoll
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany (email: Michael.Stoll@uni-bayreuth.de)

Abstract

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.

MSC classification

Secondary: 11G05: Elliptic curves over global fields 11G07: Elliptic curves over local fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 451 - 460
Copyright
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
[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) 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
[9] Serre, J.-P., Local fields (Springer, New York, 1979).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
[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