Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T16:25:15.058Z Has data issue: false hasContentIssue false
  • English
  • Français

The fine Tate–Shafarevich group

Published online by Cambridge University Press:  12 February 2007

CHRISTIAN WUTHRICH
CHRISTIAN WUTHRICH*
Affiliation:
Séction de mathématiques, CSAG, École Polytechnique Fédérale, 1015 Lausanne, Switzerland.
Get access Rights & Permissions [Opens in a new window]

Abstract

Within the Tate–Shafarevich group of an elliptic curve E defined over a number field K, there is a canonical subgroup defined by imposing stronger conditions at the places above a given prime p. This group appears naturally in the Iwasawa theory for E. We propose a study of what one can say about the relation to the full Tate–Shafarevich group. Some numerical examples are included, as well as a few conjectures.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 1 , January 2007 , pp. 1 - 12
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[CM94] Coates, J. and McConnell, G.. Iwasawa theory of modular elliptic curves of analytic rank at most 1. J. London Math. Soc. (2) 50 (1994), no. 2, 243–264.CrossRefGoogle Scholar
[Cre97] Cremona, J. E.. Algorithms for Modular Elliptic Curves, second ed. (Cambridge University Press, 1997).Google Scholar
[CS05] Coates, J. and Sujatha, R.. Fine Selmer groups of elliptic curves over p-adic Lie extensions. Math. Ann. 331 (2005), no. 4, 809–839.CrossRefGoogle Scholar
[Fla90] Flach, M.. A generalisation of the Cassels-Tate pairing. J. Reine Angew. Math. 412 (1990), 113–127.Google Scholar
[Kat04] Kato, K.. p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et application aritheoremétiques. III, Astérisque 295 (2004).Google Scholar
[Maz72] Mazur, B.. Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
[NSW00] Neukirch, J., Schmidt, A. and Wingberg, K.. Cohomology of number fields. Grund-lehren der Mathematischen Wissenschaften 323 (2000).Google Scholar
[PR95] Perrin–Riou, B.. Fonctions L p-adiques des représentations p-adiques. Asté-risque 229 (1995), 198.Google Scholar
[Rub00] Rubin, K.. Euler systems. Ann. Math. Stud. 147 (2000), Hermann Weyl Lectures. The Institute for Advanced Study.Google Scholar
[Sil96] Silverman, J. H.. The arithmetic of elliptic curves. Graduate Texts in Mathematics. vol. 99 (Springer, 1996).Google Scholar
[Tat95] Tate, J.. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki 9 (1995), pp. Exp. No. 306, 415–440.Google Scholar
[Wut04] Wuthrich, C.. The fine Selmer group and height pairings. Ph.D. Thesis (University of Cambridge, 2004).Google Scholar
[Wut05] Wuthrich, C.. Iwasawa theory of the fine Selmer group, in preparation (2005).Google Scholar