$\ell$-adic properties and congruences of $\ell$-regular partition functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916350547460096 |
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| author | El-Guindy, Ahmad Ghazy, Mostafa M. |
| author_facet | El-Guindy, Ahmad Ghazy, Mostafa M. |
| contents | We study $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode their $\ell$-adic behavior. We show that this sequence is congruent modulo increasing powers of $\ell$ to level $1$ modular forms of increasing weights. We then prove that certain $\mathbb{Z}/\ell^m\mathbb{Z}$-modules generated by our sequence are isomorphic to certain subspaces of level $1$ cusp forms of weight independent of the power of $\ell$, leading to a uniform bound on the ranks of those modules and consequently to $\ell$-adic relations between $\ell$-regular partition values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\ell$-adic properties and congruences of $\ell$-regular partition functions El-Guindy, Ahmad Ghazy, Mostafa M. Number Theory 11F33, 11P83 We study $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode their $\ell$-adic behavior. We show that this sequence is congruent modulo increasing powers of $\ell$ to level $1$ modular forms of increasing weights. We then prove that certain $\mathbb{Z}/\ell^m\mathbb{Z}$-modules generated by our sequence are isomorphic to certain subspaces of level $1$ cusp forms of weight independent of the power of $\ell$, leading to a uniform bound on the ranks of those modules and consequently to $\ell$-adic relations between $\ell$-regular partition values. |
| title | $\ell$-adic properties and congruences of $\ell$-regular partition functions |
| topic | Number Theory 11F33, 11P83 |
| url | https://arxiv.org/abs/2408.04462 |