Nonvanishing Modulo p of Hecke L-Values

  • Haruzo Hida
Part of the Springer Monographs in Mathematics book series (SMM)


We return to the setting of Sect. 6.4; thus, \(M = \mathbb{Q}[\sqrt{-D}] \subset \overline{\mathbb{Q}}\) with discriminant − D is an imaginary quadratic field in which the fixed prime (p) splits into a product of two primes \(\mathfrak{p}\overline{\mathfrak{p}}\) with \(\mathfrak{p}\neq \overline{\mathfrak{p}}\).


Prime Ideal Modular Form Elliptic Curve Eisenstein Series Ideal Class 
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  1. [AQF]
    G. Shimura, Arithmetic of Quadratic Forms. Springer Monograph in Mathematics (Springer, New York, 2010)Google Scholar
  2. [CFN]
    J. Neukirch, Class Field Theory (Springer-Verlag, Berlin, 1986)zbMATHCrossRefGoogle Scholar
  3. [EDM]
    G. Shimura, Elementary Dirichlet Series and Modular Forms. Springer Monographs in Mathematics (Springer, New York, 2007)Google Scholar
  4. [GME]
    H. Hida, Geometric Modular Forms and Elliptic Curves, 2nd edn. (World Scientific, Singapore, 2011)CrossRefGoogle Scholar
  5. [IAT]
    G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions (Princeton University Press, Princeton, NJ, 1971)zbMATHGoogle Scholar
  6. [LAP]
    H. Yoshida, Absolute CM Period. Mathematical Surveys and Monographs, vol. 106 (American Mathematical Society, Providence, RI, 2003)Google Scholar
  7. [LFE]
    H. Hida, Elementary Theory of L-Functions and Eisenstein Series. London Mathematical Society Student Texts, vol. 26 (Cambridge University Press, Cambridge, 1993)Google Scholar
  8. [Br2]
    M. Brakočević, Non-vanishing modulo p of central critical Rankin–Selberg L-values with anticyclotomic twists, preprint (posted in arXiv:1010.6066)Google Scholar
  9. [D2]
    P. Deligne, Variété abeliennes ordinaires sur un corps fini. Invent. Math. 8, 238–243 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [H04b]
    H. Hida, Non-vanishing modulo p of Hecke L-values, in Geometric Aspects of Dwork Theory, ed. by A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, F. Loeser (Walter de Gruyter, Berlin, 2004), pp. 731–780 (a preprint version posted at
  11. [H07a]
    H. Hida, Non-vanishing modulo p of Hecke L-values and application, in L-Functions and Galois Representations. London Mathematical Society Lecture Note Series, vol. 320 (Cambridge University Press, Cambridge, 2007), pp. 207–269Google Scholar
  12. [HT1]
    H. Hida, J. Tilouine, Anticyclotomic Katz p-adic L-functions and congruence modules. Ann. Sci. Éc. Norm. Sup. 4th series 26, 189–259 (1993)Google Scholar
  13. [Hs1]
    M.-L. Hsieh, On the non-vanishing of Hecke L-values modulo p. American Journal of Mathematics, 134, 1503–1539 (2012)Google Scholar
  14. [K2]
    N.M. Katz, p-Adic L-functions for CM fields. Invent. Math. 49, 199–297 (1978)Google Scholar
  15. [Sh6]
    G. Shimura, On some arithmetic properties of modular forms of one and several variables. Ann. Math. 102, 491–515 (1975) ([75c] in [CPS] II)Google Scholar
  16. [Sn2]
    W. Sinnott, On a theorem of L. Washington. Astérisque 147–148, 209–224 (1987)MathSciNetGoogle Scholar
  17. [Ws]
    L. Washington, The non-p–part of the class number in a cyclotomic \(\mathbb{Z}_{p}\)-extension. Invent. Math. 49, 87–97 (1978)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Haruzo Hida
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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