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arXiv · 1510.01202

l-adic properties of partition functions

Eva Belmont·Holden Lee·Alexandra Musat·Sarah Trebat-Leder
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Abstract

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers $p_r$ of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for $p_r$ and spt on arithmetic progressions of the form $\ell^mn+δ$ modulo powers of $\ell$. Our work gives a conceptual explanation of the exceptional congruences of $p_r$ observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

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Eva Belmont, Holden Lee, Alexandra Musat, Sarah Trebat-Leder. 2015-10-05. l-adic properties of partition functions. https://arxiv.org/abs/1510.01202

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