University of South Carolina
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Title: Congruence properties modulo prime powers for a class of partition functions
Abstract: Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1-q^{pn})^{-1}$. This function and its generalizations $p_{[c^\ell,d^m]}(n)$ are the subject of study in several recent papers. Let $\ell\geq 5$, let $j\geq 1$, and let $p\in\{2,3,5\}$. In this paper, we prove that the generating function for $p_{[1,p]}(n)$ in the progression $\beta_{p,\ell ,j}$ modulo $\ell^j$ with $24\beta_{p,\ell,j} \equiv p + 1$ (mod $\ell^j$) lies in a Hecke-invariant subspace of type $\{\eta(Dz)\eta(Dpz)F(Dz):F(z)\in M_s(\Gamma_0(p), \chi)\}$ for suitable $D \geq 1$, $s \geq 0$, and character $\chi$. When $p\in\{2,3,5\}$, we use the Hecke-invariance of these subspaces to prove that for distinct primes $\ell$ and $m \geq 5$ and $j \geq 1$, congruences of the form
\[ p_{[1,p]}\left(\frac{\ell^j m^k n + 1}{D}\right) \equiv 0 \pmod{\ell^j} \]
for all $n \geq 1$ with $m \nmid n$, where $k$ is explicitly computable and depends on the forms in the invariant subspace. This is a joint work with Matthew Boylan.