Skip to main content
SpringerLink
Log in

p-Adic congruences and modular forms of half integer weight

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Birch, B.J., Kuyk, W., eds.: Tables on elliptic curves. Modular functions of one variable IV. Lect. Notes Math. 476, 81–144. Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  2. Cohen, H.: Sums involving the values at negative integers ofL-functions of quadratic characters. Math. Ann.217, 271–285 (1975)

    Google Scholar 

  3. Cohen, H., Oesterlé, J.: Dimension des espaces de formes modulaires. Modular functions of one variable VI. Lect. Notes Math. 627, 69–78. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  4. Goldstein, C., Schappacher, N.: Séries d'Eisenstein et fonctionsL de courbes elliptiques à multiplication complexe. J. Reine Angew. Math.327, 184–218 (1981)

    Google Scholar 

  5. Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math.72, 241–265 (1983)

    Google Scholar 

  6. Gross, B.: On the factorization ofp-adicL-series. Invent. Math.57, 83–95 (1980)

    Google Scholar 

  7. Gross, B., Zagier, D.: On the critical values of HeckeL-series. Soc. Math. de France, Mémoire No. 2, 49–54 (1980)

    Google Scholar 

  8. Hardy, G.H.: On the representation of a number as the sum of any number of squares, and in particular of five or seven. Proc. Nat. Acad. Sci.4, 189–193 (1918)

    Google Scholar 

  9. Hecke, E.: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung. Math. Ann.112, 664–699 (1936). (=Math. Werke, No. 33, 591–626.)

    Google Scholar 

  10. Iwasawa, K.: Lectures onp-adicL-functions. Princeton: Princeton Univ. Press 1972

    Google Scholar 

  11. Katz, N.M.:p-adic properties of modular schemes and modular forms. Modular functions of one variable. II. Lect. Notes Math. 350, 69–190. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  12. Katz, N.M.:p-adic interpolation of real analytic Eisenstein series. Ann. Math.104, 459–571 (1976)

    Google Scholar 

  13. Katz, N.M.: The Eisenstein measure andp-adic interpolation. Am. J. Math.99, 238–311 (1977)

    Google Scholar 

  14. Koblitz, N.: Introduction to elliptic curves and modular forms. Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  15. Kohnen, W.: Kongruenzen für Hecke-Eisensteinsche Reihen reell-quadratischer Körper. Diplomarbeit, Bonn, 1977

  16. Kohnen, W.: Modular forms of half integral weight onΓ 0(4). Math. Ann.248, 249–266 (1980)

    Google Scholar 

  17. Kohnen, W.: Beziehungen zwischen Modulformen halbganzen Gewichts und Modulformen ganzen Gewichts. Bonn. Math. Schr.131, (1981)

  18. Kohnen, W.: Newforms of half-integral weight. J. Reine Angew. Math.333, 32–72 (1982)

    Google Scholar 

  19. Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann.271, 237–268 (1985)

    Google Scholar 

  20. Kohnen, W., Zagier, D.: Values ofL-series of modular forms at the center of the critical strip. Invent. Math.64, 175–198 (1981)

    Google Scholar 

  21. Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Modular forms and functions. Proc. Conf. Durham, 1983

  22. Lang, S.: Introduction to modular forms. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  23. Maeda, Y.: A congruence between modular forms of half-integral weight. Hokkaido Math. J.12, 64–73 (1983)

    Google Scholar 

  24. Manin, Yu.I.: Periods of cusp forms andp-adic Hecke series. Mat. Sb.92, 376–401 (1973)

    Google Scholar 

  25. Manin, Yu.I.: The values ofp-adic Hecke series at integers in the critical strip. Mat. Sb.93, 621–626 (1973)

    Google Scholar 

  26. Manin, Yu.I., Vishik, M.M.:p-adic Hecke series for imaginary quadratic fields. Mat. Sb.95, 357–383 (1974)

    Google Scholar 

  27. Mazur, B.: On the arithmetic of special values ofL-functions. Invent. Math.55, 207–240 (1979)

    Google Scholar 

  28. Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves. Invent. Math.25, 1–61 (1974)

    Google Scholar 

  29. Niwa, S.: Modular forms of half-integral weight and the integral of certain theta functions. Nagoya Math. J.56, 147–161 (1974)

    Google Scholar 

  30. Ogg, A.: Elliptic curves and wild ramification. Am. J. Math.89, 1–21 (1967)

    Google Scholar 

  31. Ogg, A.: Modular forms and Dirichlet series. New York: Benjamin 1969

    Google Scholar 

  32. Ramanujan, S.: Collected Papers. Cambridge: Cambridge Univ. Press 1927

    Google Scholar 

  33. Rubin, K.: Congruences for special values ofL-functions of elliptic curves with complex multiplication. Invent. Math.71, 339–364 (1983)

    Google Scholar 

  34. Serre, J.-P.: Formes modulaires et fonctions zêtap-adiques. Modular functions of one variable. III. Lect. Notes Math. 350, 191–268. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  35. Serre, J.-P., Stark, H.M.: Modular forms of weight 1/2. Modular functions of one variable. VI. Lect. Notes Math. 627, 28–67. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  36. Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton: Princeton Univ. Press 1971

    Google Scholar 

  37. Shimura, G.: On modular forms of half-integral weight. Ann. Math.97, 440–481 (1973)

    Google Scholar 

  38. Stephens, N.: Computations ofL(ψ 2k−1 D ,k). Handwritten notes and computer printout (1982)

  39. Stevens, G.: Arithmetic on modular curves. Boston: Birkhäuser 1982

    Google Scholar 

  40. Tate, J.: Local constants. Algebraic number fields (L-functions and Galois properties). Proc. L.M.S. Symp. Durham Univ. 89–131. London, New York: Academic Press 1977

    Google Scholar 

  41. Tunnell, J.: A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.72, 323–334 (1983)

    Google Scholar 

  42. Waldspurger, J.-L.: Sur les coefficients de Fourier des formes modulaires de poids demi-entier. J. Math. Pures. Appl.60, 375–484 (1981)

    Google Scholar 

  43. Weil, A.: Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.168, 149–156 (1967). (=Collected Papers, III, [1967a])

    Google Scholar 

  44. Wohlfahrt, K.: Über Operatoren Heckescher Art bei Modulformen reeller Dimension. Math. Nachr.16, 233–256 (1957)

    Google Scholar 

  45. Zagier, D.: Nombres de classes et formes modulaires de poids 3/2. C.R. Acad. Sci. Paris281, 883–886 (1975)

    Google Scholar 

  46. Zagier, D.: On the values at negative integers of the zeta function of real quadratic fields. L'Enseign. Math.22, 55–95 (1976)

    Google Scholar 

  47. Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. Modular functions of one variable. VI. Lect. Notes Math. 627, 105–169. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195, Seattle, WA, USA

    Neal Koblitz

Authors
  1. Neal Koblitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koblitz, N. p-Adic congruences and modular forms of half integer weight. Math. Ann. 274, 199–220 (1986). https://doi.org/10.1007/BF01457070

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01457070

Keywords

Search

Navigation