On Hecke modules generated by eta-products and imaginary quadratic fields (Q2832804)

The Dedekind eta-function \(\eta(\tau)\) is defined by \(\eta(\tau):= q^{1/24}\prod_{n=1}^\infty{(1-q^n)}\), where \(q=e^{2\pi{i}\tau}\), with \(\tau\) in upper half plane. The eta products \(\eta(\tau)\eta(N\tau)\) and \(\eta(\tau)^3 \eta(\lambda\tau)^3\), where \(N\) and \(\lambda\) are positive integers, \(N\equiv 23\pmod{24},\lambda\equiv 7\pmod{8}\), are cusp forms of weight 1 and weight 3, respectively. The paper proves that the dimension of the Hecke module generated by \(\eta(\tau)^3\eta(\lambda\tau)^3\) is equal to the class number of the imaginary quadratic field \(Q(\sqrt{-\lambda})\) and studies its structure as a module over the Hecke algebra. The paper also studies analogues problems with respect to Hecke module generated by \(\eta(\tau)\eta(N\tau)\).