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Weakly holomorphic modular forms of half-integral weight with nonvanishing constant terms modulo $ \ell$


Author: D. Choi
Journal: Trans. Amer. Math. Soc. 361 (2009), 3817-3828
MSC (2000): Primary 11F11, 11F33
DOI: https://doi.org/10.1090/S0002-9947-09-04708-4
Published electronically: March 4, 2009
MathSciNet review: 2491901
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Abstract: Let $ \ell$ be a prime and $ \lambda,j\geq 0$ be an integer. Suppose that $ f(z)=\sum_{n}a(n)q^n$ is a weakly holomorphic modular form of weight $ \lambda+\frac{1}{2}$ and that $ a(0)\not \equiv 0 \pmod{\ell}$. We prove that if the coefficients of $ f(z)$ are not ``well-distributed'' modulo $ \ell^j$, then

$\displaystyle \lambda=0$    or $\displaystyle 1 \pmod{\frac{\ell-1}{2}}.$

This implies that, under the additional restriction $ a(0)\not \equiv 0 \pmod{\ell}$, the following conjecture of Balog, Darmon and Ono is true: if the coefficients of a modular form of weight $ \lambda+\frac{1}{2}$ are almost (but not all) divisible by $ \ell$, then either $ \lambda\equiv 0\pmod{\frac{\ell-1}{2}}$ or $ \lambda\equiv 1 \pmod{\frac{\ell-1}{2}}$. We also prove that if $ \lambda \not \equiv 0$    and $ 1 \pmod{\frac{\ell-1}{2}},$ then there does not exist an integer $ \beta$, $ 0\leq \beta <\ell$, such that $ a(\ell n+ \beta)\equiv 0 \pmod{\ell}$ for every nonnegative integer $ n$. As an application, we study congruences for the values of the overpartition function.


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Additional Information

D. Choi
Affiliation: School of Liberal Arts and Sciences, Korea Aerospace University, 200-1, Hwajeon-dong, Goyang, Gyeonggi, 412-791, Korea
Email: choija@postech.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-09-04708-4
Received by editor(s): May 1, 2007
Received by editor(s) in revised form: August 15, 2007
Published electronically: March 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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