Congruences like Atkin's for generalized Frobenius partitions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915229425729536 |
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| author | Ahlgren, Scott Andersen, Nickolas Dicks, Robert |
| author_facet | Ahlgren, Scott Andersen, Nickolas Dicks, Robert |
| contents | In the 1960s Atkin discovered congruences modulo primes $\ell\leq 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\neq \ell$ is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes $\ell\geq 5$.
Here we consider (for primes $m\geq 5$) the $m$-colored generalized Frobenius partition functions $cϕ_m(n)$; these are natural level $m$ analogues of $p(n)$. For each such $m$ we prove that there are similar congruences for $cϕ_m(n)\pmod \ell$ for all primes $\ell$ outside of an explicit finite set depending on $m$. To prove the result we first construct, using both theoretical and computational methods, cusp forms of half-integral weight on $Γ_0(m)$ which capture the relevant values of $cϕ_m(n)$ modulo~$\ell$. We then apply previous work of the authors on the Shimura lift for modular forms with the eta multiplier together with tools from the theory of modular Galois representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruences like Atkin's for generalized Frobenius partitions Ahlgren, Scott Andersen, Nickolas Dicks, Robert Number Theory In the 1960s Atkin discovered congruences modulo primes $\ell\leq 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\neq \ell$ is prime. Recent work of the first author with Allen and Tang shows that such congruences exist for all primes $\ell\geq 5$. Here we consider (for primes $m\geq 5$) the $m$-colored generalized Frobenius partition functions $cϕ_m(n)$; these are natural level $m$ analogues of $p(n)$. For each such $m$ we prove that there are similar congruences for $cϕ_m(n)\pmod \ell$ for all primes $\ell$ outside of an explicit finite set depending on $m$. To prove the result we first construct, using both theoretical and computational methods, cusp forms of half-integral weight on $Γ_0(m)$ which capture the relevant values of $cϕ_m(n)$ modulo~$\ell$. We then apply previous work of the authors on the Shimura lift for modular forms with the eta multiplier together with tools from the theory of modular Galois representations. |
| title | Congruences like Atkin's for generalized Frobenius partitions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.03954 |