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arXiv · 1706.07453

Congruences for a mock modular form on $\operatorname{SL}_2(\mathbb{Z})$ and the smallest parts function

Scott Ahlgren·Byungchan Kim
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Abstract

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight $3/2$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo $\ell$. As another application, we show that congruences modulo $\ell$ for the smallest parts function are rare in a precise sense.

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Scott Ahlgren, Byungchan Kim. 2017-06-22. Congruences for a mock modular form on $\operatorname{SL}_2(\mathbb{Z})$ and the smallest parts function. https://arxiv.org/abs/1706.07453

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