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manuscripta mathematica

, Volume 151, Issue 3–4, pp 493–504 | Cite as

The special L-value of the winding quotient of square-free level

  • Amod AgasheEmail author
Article
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Abstract

We give an explicit formula that expresses the algebraic part of the special L-value of the winding quotient of square-free level as a rational number, and interpret it in terms of the Birch–Swinnerton-Dyer conjecture.

Mathematics Subject Classification

11G10 11G18 11G40 

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References

  1. 1.
    Agashe, A.: On invisible elements of the Tate–Shafarevich group. C. R. Acad. Sci. Paris Sér. I Math. 328(5), 369–374 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agashe, A.: The Birch and Swinnerton-Dyer Formula for Modular Abelian Varieties of Analytic Rank Zero, Ph.D. thesis, University of California, Berkeley (2000). http://www.math.fsu.edu/~agashe/math.html
  3. 3.
    Agashe, A.: A visible factor of the special L-value. J. Reine Angew. Math. 644, 159–187 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Agashe, A., Ribet, K., Stein, W.A.: The Manin constant. Pure Appl. Math. Q. 2(2), 617–636 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Agashe, A., Stein, W.: Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero. Math. Comput. 74(249), 455–484 (electronic). With an appendix by J. Cremona and B. Mazur (2005)Google Scholar
  6. 6.
    Atiyah, M.F., Wall, C.T.C.: Cohomology of groups. In: Algebraic Number Theory (Proceedings Instructional Conferences, Brighton, 1965), pp. 94–115. Thompson, Washington, DC (1967)Google Scholar
  7. 7.
    Banerjee, D., Krishnamoorthy, S.: The Eisenstein Elements Inside the Space of Modular Symbols, preprint. https://sites.google.com/site/mathsban/preprints
  8. 8.
    Bourbaki, N.: Elements of Mathematics. General Topology. Part 1. Hermann, Paris (1966)zbMATHGoogle Scholar
  9. 9.
    Cremona, J.E.: Algorithms for Modular Elliptic Curves, 2nd edn. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  10. 10.
    Edixhoven, B.: Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one. J. Inst. Math. Jussieu 5(1), 1–34 (2006). With appendix A (in French) by Jean-François Mestre and appendix B by Gabor WieseGoogle Scholar
  11. 11.
    Emerton, M.: Optimal quotients of modular Jacobians. Math. Ann. 327(3), 429–458 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Joyce, A.: The Manin constant of an optimal quotient of \(J_0(431)\). J. Number Theory 110(2), 325–330 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kolyvagin, V.A., Logachev, D.Y.: Finiteness of the Shafarevich–Tate group and the group of rational points for some modular abelian varieties. Algebra Anal. 1(5), 171–196 (1989)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lang, S.: Number Theory. III. Springer, Berlin (1991). Diophantine geometryGoogle Scholar
  15. 15.
    Mazur, B.: Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47, 33–186 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124(1–3), 437–449 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Merel, L.: L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de \({J}_{0}({p})\). J. Reine Angew. Math. 477, 71–115 (1996)MathSciNetGoogle Scholar
  18. 18.
    Parent, P.: Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew. Math. 506, 85–116 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stein, W.A.: The Cuspidal Subgroup of \(J_0(N)\), http://modular.math.washington.edu/tables/cuspgroup/index.html

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA

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