, Volume 98, Issue 1, pp 25-36
Date: 24 Dec 2011

A mod Atkin–Lehner theorem and applications

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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.

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