Non-zero coefficients of half-integral weight modular forms mod $\ell$

Joël Bellaïche; Ben Green; Kannan Soundararajan

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arXiv · 1704.07440

Non-zero coefficients of half-integral weight modular forms mod $\ell$

Joël Bellaïche·Ben Green·Kannan Soundararajan

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Abstract

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function $p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the $j$-invariant, is odd.

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Joël Bellaïche, Ben Green, Kannan Soundararajan. 2017-04-24. Non-zero coefficients of half-integral weight modular forms mod $\ell$. https://arxiv.org/abs/1704.07440

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