Ramanujan's partition generating functions modulo ℓ

For the partition function p(n), Ramanujan proved the striking identities $\begin{array}{c}\begin{array}{cc}{P}_5\left(q\right):=\sum _{n\ge 0}p(5n+4)q^n& =5\prod _{n\ge 1}\frac{{\left(q^5,;,q^5\right)}_{\infty }^5}{{(q;q)}_{\infty }^6},\\ P_7\left(q\right):=\sum _{n\ge 0}p(7n+5)q^n& =7\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^3}{{(q;q)}_{\infty }^4}+49q\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^7}{{(q;q)}_{\infty }^8},\end{array}\end{array}$ where ${(q;q)}_{\infty }:={\prod }_{n\ge 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes $\ell \ge 5,$ closed form expressions of the power series $P_{\ell }\left(q\right):=\sum _{n\ge 0}p(\ell n-{\delta }_{\ell })q^n(mod\ell ),$ where ${\delta }_{\ell }:=\frac{{\ell }^2-1}{24}.$ In this paper, we prove that $P_{\ell }\left(q\right)\equiv c_{\ell }\frac{T_{\ell }\left(q\right)}{{\left(q^{\ell },;,q^{\ell }\right)}_{\infty }}(mod\ell ),$ where $c_{\ell }\in Z$ is explicit and $T_{\ell }\left(q\right)$ is the generating function for the Hecke traces of $\ell$ -ramified values of special Dirichlet series for weight $\ell -1$ cusp forms on ${\text{SL}}_2\left(Z\right)$ . This is a new proof of Ramanujan's congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10.