Advertisement

Special Values of L-functions

  • Ze-Li Dou
  • Qiao Zhang

Abstract

In the last chapter, we studied the analytic properties of L-functions, in particular their zero distributions. If we simply regard an L-function as a complex analytic function, then its values at different points seem to be equally important or unimportant. However, since our L-functions all come from arithmetic geometric or automorphic origins, it turns out that the function values at certain points are more transparent in revealing the associated arithmetic or automorphic information, and thus deserve special attention. It is for this reason that such L-function values are called special values.

Keywords

Functional Equation Elliptic Curve Elliptic Curf Class Number Analytic Number Theory 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ba66]
    Baker, A. “Linear forms in the logarithms of algebraic numbers.” Mathematika 13 (1966): 204–216.CrossRefGoogle Scholar
  2. [Ba71]
    Baker, A. “Imaginary quadratic fields with class number 2.” Annals of Math. 94 (1971): 139–152.CrossRefzbMATHGoogle Scholar
  3. [CF00]
    Conrey, J., Farmer, D. “Mean values of L-functions and symmetry.” Internat. Math. Res. Notices 17 (2000): 883–908.MathSciNetCrossRefGoogle Scholar
  4. [Deu33]
    Deuring, M. “Imaginär-quadratische Zahlkörper mit der Klassenzahl (1).” Math. Z. 37 (1933): 405–415.MathSciNetCrossRefGoogle Scholar
  5. [DGH03]
    Diaconu, A., Goldfeld, D., Hoffstein, J. “Multiple Dirichlet series and moments of zeta and L-functions.” Compositio Math. 139 (2003): 297–360.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Du88]
    Duke, W. “Hyperbolic distribution problems and half-integral weight Maass forms.” Invent. Math. 92 (1988): 73–90.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Go76]
    Goldfeld, D. “The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer.” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976): 624–663.MathSciNetGoogle Scholar
  8. [Goo84]
    Good, A. “The convolution method for Dirichlet series.” In The Selberg trace formula and related topics (Brunswick, Maine, 1984), American Mathematical Society, Providence, 1986: 207–214.CrossRefGoogle Scholar
  9. [GZ86]
    Gross, B., Zagier, D. “Heegner points and derivatives of L-series.” Invent. Math. 84 (1986): 225–320.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [He52]
    Heegner, K. “Diophantische analysis und modulfunktionen.” Math. Z. 56 (1952): 227–153.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Hei34]
    Heilbronn, H. “On the class number in imaginary quadratic fields.” Quart. J. Math. Oxford (2) 5 (1934): 150–160.CrossRefzbMATHGoogle Scholar
  12. [HL34]
    Heilbronn, H., Linfoot, E. “On the imaginary quadratic corpora of class number one.” Quart. J. Math. Oxford (2) 5 (1934): 293–301.CrossRefzbMATHGoogle Scholar
  13. [IK04]
    Iwaniec, H., Kowalski, E. Analytic number theory. American Mathematical Society, 2004.Google Scholar
  14. [La18]
    Landau, E. “Über die Klasszahl imaginär-quadratischer Zahlkörper.” Göttinger Nachr. (1918): 285–295.Google Scholar
  15. [St67]
    Stark, H. “A complete determination of the complex quadratic fields of classumber one.” Mich. Math. J. 14 (1967): 1–27CrossRefzbMATHGoogle Scholar
  16. [St71]
    Stark, H. “A transcendence theorem for classumber problems I, II.” Annals of Math. (2) 94 (1971): 153–173; ibid. 96 (1972): 174–209.CrossRefzbMATHGoogle Scholar
  17. [Wi95]
    Wiles, A. “Modular elliptic curves and Fermat’s last theorem.” Ann. of Math. (2) 141 (1995): 443–551.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [Za84]
    Zagier, D. “L-series of elliptic curves, the Birch-Swinnerton-Dyer conjecture, and the class number problem of Gauss.” Notices Amer. Math. Soc. 31 (1984): 739–743.MathSciNetzbMATHGoogle Scholar
  19. [Zh05a]
    Zhang, Q. “On the cubic moment of quadratic Dirichlet L-functions.” Math. Res. Lett. 12 (2005): 413–424.MathSciNetzbMATHGoogle Scholar
  20. [Zh05b]
    Zhang, Q. “Integral mean values of modular L-functions.” J. Number Theory 115 (2005): 100–122.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Zh06a]
    Zhang, Q. “Applications of multiple Dirichlet series in mean values of Lfunctions.” In Multiple Dirichlet series, automorphic forms, and analytic number theory, American Mathematical Society, 2006: 43–57.Google Scholar
  22. [Zh06b]
    Zhang, Q. “Integral mean values of Maass L-functions.” Int. Math. Res. Not. 2006, Art. ID 41417, 19 pp.Google Scholar

Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ze-Li Dou
    • 1
  • Qiao Zhang
    • 1
  1. 1.Department of MathematicsTexas Christian UniversityFort WorthUSA

Personalised recommendations