Ramanujan type congruences for quotients of level 7 Klein forms (Q1998895)

Let \(a_t(n)\) denote the number of \(t\)-core partitions of \(n\). Motivated by congruences modulo powers of \(5, 7, 11\) for partition function \(p(n)\) due to Ramanujan, \textit{F. Garvan} et al. [Invent. Math. 101, No. 1, 1--17 (1990; Zbl 0721.11039)] derived the following Ramanujan type congruences for \(a_t(n)\) with \(t\in\{5,7,11\}\): \[ a_t{\left(t^kn-\delta_t\right)}\equiv0\pmod{t^k},\qquad\delta_t=(t^2-1)/24. \] The authors investigate a certain colored-weighted partition function \(P_{a_0,a_1,a_2,a_3}(n)\), defined by \[ \sum_{n=0}^\infty P_{a_0,a_1,a_2,a_3}(n)q^n=(q^7;q^7)_\infty^{a_0}(q,q^6;q^7)_\infty^{a_1} (q^2,q^5;q^7)_\infty^{a_2}(q^3,q^4;q^7)_\infty^{a_3}, \] where \[ (a;q)_\infty=\prod_{j=0}^\infty(1-aq^j)\qquad\text{and}\qquad(a_1,\ldots,a_k;q)_\infty =\prod_{j=1}^k(a_j;q)_\infty. \] Note that \(a_7(n)=P_{6,-1,-1,-1}\). In this paper under review, the author provide a unified derivation of Ramanujan type congruences modulo powers of 7 for \(P_{a_0,a_1,a_2,a_3}(n)\) with \((a_0,a_1+a_2+a_3)\in\{(4,-2),(6,-3)\}\). The main technique is to construct generators for the graded algebra of holomorphic modular forms for \(\varGamma(7)\) via Klein forms. For example, the authors prove that for any \(n\geq1\) and \(k\geq1\), \begin{align*} P_{6,1,0,-4}(7^kn-3) &\equiv0\pmod{7^{2k}},\\ P_{6,4,-2,-5}(7^kn-4) &\equiv0\pmod{7^k},\\ P_{4,1,-1,-2}(7^kn-2) &\equiv0\pmod{7^k}. \end{align*} On the other hand, let \(Q_k(n)\) denote the number of representations of \(n=x_1+x_2+\cdots+x_k\), where \(x_i=m_i^2+m_in_i+2n_i^2\) \((1\leq i\leq k)\) and \(m_i, n_i\in\mathbb{Z}\). The authors also establish a wealth of congruences for \(Q_k(n)\). For instance, for any \(n\geq0\), \[ Q_4(7n+\ell) \equiv\begin{cases} 0\pmod{8},\;&\text{if }\left(\frac{\ell}{7}\right)=1,\\ 0\pmod{16},\;&\text{if }\left(\frac{\ell}{7}\right)=-1. \end{cases} \] For any odd prime \(k\) and \(n\geq0\), if \(\left(\frac{r}{7}\right)=-1\) and \(\left(\frac{k}{7}\right)=1\), then \[ Q_k(7n+k)\equiv0\pmod{8k}, \] where \(\left(\frac{\cdot}{\cdot}\right)\) is the Legendre symbol.