Skip to main content
SpringerLink
Log in

Coefficients of half-integral weight modular forms modulo ℓj

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

Suppose that ℓ≥5 is prime, that j≥0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are “well-distributed” modulo ℓj. As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power ℓj with ℓ≥5, that the ordinary partition function p(n) takes each value modulo ℓj infinitely often.

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

Similar content being viewed by others

References

  1. Ahlgren, S.: The partition function modulo composite integers M. Math. Ann. 318, 795–803 (2000)

    MATH  Google Scholar 

  2. Ahlgren, S., Boylan, M.: Arithmetic properties of the partition function. Invent. Math. 153(3), 487–502 (2003)

    MATH  Google Scholar 

  3. Ahlgren, S., Ono, K.: Congruence properties for the partition function. Proc. Nat. Acad. Sci. 98(23), 12882–12884 (2001)

    MATH  Google Scholar 

  4. Ahlgren, S., Ono, K.: Congruences and conjectures for the partition function. Contemp. Math. 291, 1–10 (2001)

    MATH  Google Scholar 

  5. Atkin, A. O. L.: Multiplicative congruence properties and density problems for p(n). Proc. London Math. Soc. (3) 18, 563–576 (1968)

  6. Balog, A., Darmon, H., Ono, K.: Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions. Proceedings for a conference in honor of Heini Halberstam, vol. 1 Birkhäuser Boston, MA 1996 pp. 105–127

  7. Bruinier, J.: Non-vanishing modulo ℓ of Fourier coefficients of half-integral weight modular forms. Duke Math. J. 98, 595–611 (1999)

    MATH  Google Scholar 

  8. Bruinier, J., Ono, K.: Fourier coefficients of half-integral weight modular forms. J. Number Th. 99, 164–179 (2003)

    MATH  Google Scholar 

  9. Bruinier, J., Ono, K.: Fourier coefficients of half-integral weight modular forms (corrigendum). J. Number Th. 104, 378–379 (2004)

    MATH  Google Scholar 

  10. Deligne, P., Serre, J.-P.: Formes modulaires de poids 1. Ann. Sci. École Normale Sup. 4e sér. 7, 507–530(1974)

    Google Scholar 

  11. Gross, B.H.: A tameness criterion for Galois representations attached to modular forms (mod p). Duke Math. J. 61(2), 445–517 (1990)

    MATH  Google Scholar 

  12. Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York 1980

  13. Hida, H.: Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts, 26, Cambridge University Press, 1993

  14. Katz, N.: A result on modular forms in characteristic p. Lecture Notes in Math. vol. 601 (Modular functions of one variable, V)(1977), Springer Verlag Berlin, pp. 53–61

  15. Kløve, T.: Recurrence formulae for the coefficients of modular forms and congruences for the partition function and for the coefficients of j(τ), (j(τ)-1728)1/2, and j(τ)1/3. Math. Scand. 23, 133–159 (1969)

    Google Scholar 

  16. Koblitz, N.: Introduction to elliptic curves and modular forms. Springer-Verlag, New York, Graduate Texts in Mathematics, No. 97, 1984

  17. Kolberg, O.: Note on the parity of the partition function. Math. Scand. 7, 377–378 (1959)

    MATH  Google Scholar 

  18. Newman, M.: Periodicity modulo m and divisibility properties of the partition function. Trans. Amer. Math. Soc. 97, 225–236 (1960)

    MATH  Google Scholar 

  19. Ono, K.: Distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000)

    MATH  Google Scholar 

  20. Ribet, K.: On ℓ-adic representations attached to modular forms. Invent. Math. 28, 245–275 (1975)

    MATH  Google Scholar 

  21. Ribet, K.: On ℓ-adic representations attached to modular forms, II. Glasgow Math. J. 27, 185–194 (1985)

    MATH  Google Scholar 

  22. Serre, J.-P.: Congruences et formes modulaires (d’après Swinnerton-Dyer). Sém. Bourbaki 416, 319–338 (1971–1972)

    Google Scholar 

  23. Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. Swinnerton-Dyer, H. P. F.: On ℓ-adic representations and congruences for modular forms. Lecture Notes in Math., vol. 350 (Modular functions of one variable III)(1973), Springer-Verlag Berlin, pp. 1–55

  26. Vignéras, M. F.: Facteurs gamma et équations fonctionelles Lecture Notes in Math. vol. 627 (Modular functions of one variable VI)(1977), Springer-Verlag Berlin pp. 79–103

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Scott Ahlgren & Matthew Boylan

Authors
  1. Scott Ahlgren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew Boylan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott Ahlgren.

Additional information

Mathematics Subject Classification (2000): 11F33, 11P83

The first author thanks the National Science Foundation for its support through grant DMS 01-34577. The second author thanks the National Science Foundation for its support through a VIGRE postdoctoral fellowship.

Acknowledgement The authors thank the referee for suggestions which improved the exposition in this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ahlgren, S., Boylan, M. Coefficients of half-integral weight modular forms modulo ℓj. Math. Ann. 331, 219–239 (2005). https://doi.org/10.1007/s00208-004-0555-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0555-9

Keywords

Search

Navigation