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A mod ℓ Atkin–Lehner theorem and applications

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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 ℓ. We show that the same conclusion holds if gcd(N ℓ,m) = 1 and the nebentypus character is trivial at ℓ. We use this to study the parity of the partition function and the coefficients of Klein’s j-function.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Ken Ono

  2. Department of Mathematics, DePaul University, Chicago, IL, 60614, USA

    Nick Ramsey

Authors
  1. Ken Ono
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  2. Nick Ramsey
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Correspondence to Ken Ono.

Additional information

The first author is grateful for support from the NSF and the Asa Griggs Candler Fund.

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Ono, K., Ramsey, N. A mod ℓ Atkin–Lehner theorem and applications. Arch. Math. 98, 25–36 (2012). https://doi.org/10.1007/s00013-011-0347-x

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