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On the Proportion of Quadratic Twists for Non-vanishing and Vanishing Central Values of L-Functions Attached to Primitive Forms

  • Kenji MakiyamaEmail author
Conference paper
  • 642 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 115)

Abstract

Let \(S_{2k}^{\mathrm{new}}(\Gamma _{0}(N))\) be the space of newforms of weight 2k on \(\Gamma _{0}(N)\). Let k and N be positive integers and \(f \in S_{2k}^{\mathrm{new}}(\Gamma _{0}(N))\) a primitive form. Let r be either 0 or 1.

Keywords

Primitive Form Newform Half-integral Weight Cusp Forms Shimura Correspondence Semistable Elliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The author is very grateful to Professor Atsushi Yamagami for variable guidance and kind help. He is also grateful to Professor Atsushi Murase and Professor Bernhard Heim for valuable comments on improving the manuscript and supports for my participation in the international workshop on mathematics held at Oman in February 2012.

References

  1. 1.
    C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over \(\mathbb{Q}\), or wild 3-adic exercises. J. Amer. Math. Soc. 14(4), 843–939 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields II. Proc. Roy. Soc. Lond. A 322, 405–420 (2001)Google Scholar
  3. 3.
    D.W. Farmer, K. James, The irreducibility of some level 1 Hecke polynomials. Math. Comp. 71(239), 1263–1270 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Goldfeld, Conjectures on elliptic curves over quadratic fields. Number Theory, Carbondale, Springer Lect. Notes. 751, 108–118 (1979)Google Scholar
  5. 5.
    H. Hida, Y. Maeda, Non-abelian base change for totally real fields. Olga Taussky-Todd: in memoriam. Pacific J. Math. Special Issue, 189–217 (1997)Google Scholar
  6. 6.
    K. James, L-series with non-zero central critical value. J. AMS 11(3), 635–641 (1998)zbMATHGoogle Scholar
  7. 7.
    W. Kohnen, On the proportion of quadratic character twists of L-functions attached to cusp forms not vanishing at the central point. J. Rein. Angew. Math. 508, 179–187 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    W. Kohnen, D. Zagier, Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64, 175–198 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J. Nakagawa, K. Horie, Elliptic curves with no rational points. Proc. AMS 104(1), 20–24 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, vol. 102 (American Mathematical Society, 2004)Google Scholar
  11. 11.
    K. Ono, C. Skinner, Nonvanishing of quadratic twists of modular L-function. Invent. Math. 134, 651–660 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    A. Perelli, J. Pomykala, Averages of twisted L-functions. Acta Arith. 80, 149–163 (1997)zbMATHMathSciNetGoogle Scholar
  13. 13.
    H. Sakata, On the Kohnen-Zagier formula in the general case of ‘4× general odd’ level. Nagoya Math. J. 190, 63–85 (2008)zbMATHMathSciNetGoogle Scholar
  14. 14.
    V. Vatsal, Canonical periods and congruence formulae. Duke Math. J. 98, 397–419 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    V. Vatsal, Rank-one twists of a certain elliptic curve. Math. Ann. 311, 791–794 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures et Appl. 60, 375–484 (1997)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsKyoto Sangyo UniversityKyotoJapan

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