Archiv der Mathematik

, Volume 98, Issue 1, pp 25–36 | Cite as

A mod Atkin–Lehner theorem and applications



If f(z) is a weight \({k\in \frac{1}{2}\mathbb {Z}}\) meromorphic modular form on Γ0(N) satisfying
$$f(z)=\sum_{n\geq n_0} a_ne^{2\pi i mnz}, $$
where \({m \nmid N,}\) then f is constant. If k ≠ 0, then f = 0. Atkin and Lehner [2] derived the theory of integer weight newforms from this fact. We use the geometric theory of modular forms to prove the analog of this fact for modular forms modulo . We show that the same conclusion holds if gcd(N ,m) = 1 and the nebentypus character is trivial at . We use this to study the parity of the partition function and the coefficients of Klein’s j-function.


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  1. 1.
    C. Alfes, Parity of the coefficients of Klein’s j-function, Proc. Amer. Math. Soc., in press.Google Scholar
  2. 2.
    A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), pages 134–160.Google Scholar
  3. 3.
    Bruinier J. H., Ono K.: Heegner divisors, L-functions, and harmonic weak Maass forms. Ann. Math 172, 2135–2181 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Conrad B.: Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu 6, 209–278 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    N. M. Katz, p-adic properties of modular schemes and modular forms, in Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 69–190. Lecture Notes in Mathematics, Vol. 350. Springer, Berlin, 1973.Google Scholar
  6. 6.
    N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, volume 108 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1985.Google Scholar
  7. 7.
    Mazur B.: Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. No 47, 33–186 (1977)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Ono K.: Parity of the partition function in arithmetic progressions. J. Reine Angew. Math 472, 1–15 (1996)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ono K.: The parity of the partition function. Adv. Math. 225, 349–366 (2010)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Parkin T.R., Shanks D.: On the distribution of parity in the partition function. Math. Comp 21, 466–480 (1967)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    S. Radu, A proof of Subbarao’s conjecture, J. reine Angew. Math., in press.Google Scholar
  12. 12.
    Subbarao M.: Some remarks on the partition function. Amer. Math. Monthly 73, 851–854 (1966)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    D. Zagier, Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I. Intl. Press, Somerville (2002), 211–244.Google Scholar

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.Department of MathematicsDePaul UniversityChicagoUSA

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