Archiv der Mathematik

, Volume 98, Issue 1, pp 25–36

A mod Atkin–Lehner theorem and applications

Article

DOI: 10.1007/s00013-011-0347-x

Cite this article as:
Ono, K. & Ramsey, N. Arch. Math. (2012) 98: 25. doi:10.1007/s00013-011-0347-x

Abstract

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.

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

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