Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T08:28:15.386Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristic elements, pairings and functional equations over the false Tate curve extension

Published online by Cambridge University Press:  01 May 2008

GERGELY ZÁBRÁDI
GERGELY ZÁBRÁDI*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB.
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a prime p≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 535 - 574
Copyright
Copyright © Cambridge Philosophical Society 2008

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

[1]Ardakov, K. and Wadsley, S.. Characteristic elements for p-torsion Iwasawa modules. J. Algebraic Geom. 15(2006), 339–377.CrossRefGoogle Scholar
[2]Bouganis, Th. and Dokchitser, V.. Algebraicity of L-values for elliptic curves in a false Tate curve tower, preprint.Google Scholar
[3]Coates, J., Fukaya, T., Kato, K. and Sujatha, R.. Rootnumbers, Selmer groups, and non-commutative Iwasawa theory, in preparation.Google Scholar
[4]Coates, J., Fukaya, T., Kato, K., Sujatha, R. and Venjakob, O.. The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208.CrossRefGoogle Scholar
[5]Coates, J., Schneider, P. and Sujatha, R.. Links between cyclotomic and GL2 Iwasawa theory, Documenta Mathematica. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 187–215.CrossRefGoogle Scholar
[6]Coates, J., Schneider, P. and Sujatha, R.. Modules over Iwasawa algebras. J. Inst. Math. Jussieu 2 (1) (2003), 73–108.CrossRefGoogle Scholar
[7]Cremona, J.. Elliptic curves data http://www.maths.nottingham.ac.uk/personal/jec/ftp/data.Google Scholar
[8]Darmon, H. and Tian, Y.. Heegner points over false Tate curve extensions. Talk in Montreal (2005).Google Scholar
[9]Deligne, P.. Valeur de fonctions L et périodes d'intégrales. Proc. Sympos. Pure Math. 33 Part 2, (1979), 313–346.Google Scholar
[10]Dokchitser, V. (with an appendix by T. Fisher). Root numbers of non-abelian twists of elliptic curves. Proc. London Math. Soc. (3) 91 (2005), 300–324.Google Scholar
[11]Dokchitser, T. and Dokchitser, V.. (with an appendix by J. Coates and R. Sujatha). Computations in non-commutative Iwasawa theory, preprint.Google Scholar
[12]Flach, M.. A generalisation of the Cassels–Tate pairing. J. Reine Angew. Math. 412 (1990), 113–127.Google Scholar
[13]Fukaya, T. and Kato, K.. A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, preprint.Google Scholar
[14]Greenberg, R.. Iwasawa theory for p-adic representations, in Algebraic number theory. Adv. Stud. Pure Math. 17 (1989), 97–137.CrossRefGoogle Scholar
[15]Greenberg, R.. Introduction to Iwasawa theory for elliptic curves, in Arithmetic algebraic geometry (Park City, UT, 1999), 407–464.Google Scholar
[16]Hachimori, Y. and Matsuno, K.. An analogue of Kida's formula for the Selmer groups of elliptic curves. J. Algebraic Geom. 8 (1999), 581–601.Google Scholar
[17]Hachimori, Y. and Venjakob, O.. Completely faithful Selmer groups over Kummer extensions, Documenta Mathematica. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 443–478.Google Scholar
[18]Jannsen, U.. Iwasawa modules up to isomorphism, in Algebraic number theory. Adv. Stud. Pure Math. 17 (1989), 171–207.CrossRefGoogle Scholar
[19]Kato, K.. K1 of some non-commutative completed group rings. K-Theory 34 (2005), no. 2, 99–140.Google Scholar
[20]Nekovář, J.. On the parity of ranks of Selmer groups. II, C. R. Acad. Sci. Paris Sr. I Math. 332 (2001), no. 2, 99–104.Google Scholar
[21]Perrin–Riou, B.. Groupes de Selmer et accouplements; Cas particulier des courbes elliptiques, Documenta Mathematica. Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 725–760.CrossRefGoogle Scholar
[22]Vaserstein, L. N.. On the Whitehead determinant for semi-local rings. J. Algebra 283 (2005), no. 2, 690–699.CrossRefGoogle Scholar
[23]Venjakob, O. (with an appendix by D. Vogel), A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory. J. Reine Angew. Math. 559 (2003), 153–191.Google Scholar
[24]Vogel, D.. Nonprincipal reflexive left ideals in Iwasawa algebras II, preprint, http://homepages.uni-regensburg.de/vod05208/nonprincipal2.pdfGoogle Scholar