Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 475–484 | Cite as

A Stickelberger index for the Tate-Shafarevich group revisited

  • Petra ForsterEmail author
  • Hendrik Kasten


We obtain an algebraic description of the critical value of the L-series of an elliptic curve of rank zero over an abelian number field with conductor coprime to the one of the curve. We do this by construction of matching Stickelberger-ideals and computation of their indices. By inserting the result into the formula of the conjecture of Birch and Swinnerton-Dyer we can describe the order of the Tate-Shafarevich group by a number of algebraic expressions.

Mathematics Subject Classification (2000)

11G05 11F11 


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  1. 1.
    Bourbaki N.: Commutative Algebra. Hermann, Paris (1972)zbMATHGoogle Scholar
  2. 2.
    Kasten H.: A Stickelberger index for the Tate-Shafarevich group. manuscripta math. no. 117/4, 511–523 (2005)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Manin Y.: Parabolic Points and Zeta-functions of Modular Curves. Math. USSR Izvestija no. 6–1, 19–64 (1972)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publications Mathématiques, no. 47. I.H.E.S., (1977)Google Scholar
  5. 5.
    Tate J.: On the conjectures of BSD and a geometric analog. Séminaire Bourbaki no. 306, 1–26 (1966)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut für Algebra und GeometrieUniversität KarlsruheKarlsruheGermany
  2. 2.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany

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