A refinement of Ramanujan's congruences modulo powers of 7 and 11 (Q2890243)

Let \(p(n)\) denote the number of partitions of \(n\). Recall the partition congruences NEWLINE\[NEWLINE p(7^j n + \beta_7(j)) \equiv 0 \pmod{7^{\lfloor j/2 \rfloor +1}} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE p(11^j n + \beta_{11}(j)) \equiv 0 \pmod{11^{j}}, NEWLINE\]NEWLINE where \(\beta_m(j) := 1/24 \pmod{m^j}\). In proving the congruences modulo powers of \(7\), Watson showed that NEWLINE\[NEWLINE \frac{1}{5^{\lfloor j/2 \rfloor} 7^{\lfloor j/2 \rfloor+1}}\sum_{n \geq 0} p(7^jn+\beta_7(j))q^{24n+17} \equiv \eta^{17}(24z) \pmod{7}, NEWLINE\]NEWLINE where \(q := e^{2 \pi i z}\) and \(\eta(z)\) is the usual Dedekind \(\eta\)-function. By work of Newman, \(\eta^{17}(24z)\) is an eigenform for the Hecke operators \(T(p^2)\), and here the author uses the Shimura correspondence and the theory of modular forms mod \(\ell\) to express the eigenvalues modulo \(\ell\) in terms of \(\#E_7(\mathbb{F}_{\ell})\), where \(E_7\) is the elliptic curve \(y^2+xy+y = x^3+x^2-4x+5\). This leads to congruences like the following: Suppose that \(j \geq 1\) is odd, that \(\#E_7(\mathbb{F}_{\ell}) \equiv 1+\ell \pmod{7}\), and that \(0 \leq r,s \leq \ell-1\) are integers such that \(24r+17 \equiv 0 \pmod{\ell}\) and \(24(r+s\ell) \neq 0 \pmod{\ell^2}\). Then for all \(N \geq 0\) one has NEWLINE\[NEWLINE p(7^j\ell^2(\ell^2N+\ell s +r)+\beta_7(j,\ell)) \equiv 0 \pmod{7^{\lfloor j/2\rfloor+2}}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \beta_7(j,\ell) := \frac{17 \cdot 7^j \cdot \ell^2+1}{24}. NEWLINE\]NEWLINE In other words, one has subprogressions of \(7^j n + \beta_7(j)\) for which the partition congruence modulo \(7^{\lfloor j/2 \rfloor +1}\) is in fact a congruence modulo \(7^{\lfloor j/2 \rfloor +2}\). Using results of Serre on Galois representations, the author shows that the proportion of primes \(\ell\) for which \(\#E_7(\mathbb{F}_{\ell}) \equiv 1+\ell \pmod{7}\) is \(\frac{7}{48}\). There are similar congruences when \(\#E_7(\mathbb{F}_{\ell}) \equiv 1+\ell \pm 1 \pmod{7}\) and/or when \(j\) is even, and analogous results hold modulo powers of \(11\).