Hecke operators on rational period functions on the Hecke groups (Q1320072)

The groups in question here are \(G(\sqrt {2})\) and \(G(\sqrt {3})\), the two Hecke groups commensurable with the modular group \(\Gamma(1)= G(1)\). (\(G(\lambda)\) denotes the group of linear fractional transformations generated by \(S_ \lambda: z\to z+\lambda\), and \(T: z\to -1/z\).) A modular integral (MI) on \(G(\lambda)\) of weight \(zk\), \(k\in \mathbb Z\), is a function \(f\) holomorphic in the upper half-plane, such that \(f(z+ \lambda)= f(z)\), \(z^{-2k} f(-1/z)= f(z)+ q(z)\), where \(q\) is a rational function. \(q\) is called the rational period function (RPF) associated with \(f\). (If \(q=0\) then \(f\) is a modular form on \(G(\lambda)\).) \textit{J. Bogo} and \textit{W. Kuyk} [J. Algebra 43, 585--605 (1976; Zbl 0353.10019)] have introduced Hecke operators \(T_ \lambda (n)\) for the groups \(G(\lambda)\), \(\lambda= \sqrt{2}\) and \(\sqrt{3}\). Natural analogues of the usual Hecke operators \(T(n)\) on \(\Gamma(1)\), the \(T_ \lambda (n)\) operate on the space of modular forms and the subspace of cusp forms, on \(G(\lambda)\). In [Duke Math. J. 45, 47-62 (1978; Zbl 0374.10014)] the reviewer studied the effect of the \(T(n)\) on the MI's with respect to \(\Gamma(1)\); in acting upon MI's the \(T(n)\) induce operators \(\hat{T}(n)\) acting on the space of RPF's for \(\Gamma(1)\). The same procedure can be applied to the MI's (and, consequently, to the RPF's) connected with \(G (\sqrt{2})\) and \(G(\sqrt {3})\), by bringing to bear the operators \(T_ \lambda (n)\). In [Contemp. Math. 143, 89--108 (1993; Zbl 0790.11044)] the author and \textit{D. Zagier} demonstrated how to define \(\hat{T} (n)\) on the RPF's for \(\Gamma(1)\) directly from the algebraic properties of \(\Gamma(1)\), that is, without introducing the MI's. The purpose of the present article is to carry over this algebraic construction of Hecke operators on RPF's to \(\hat {T}_ \lambda(n)\), with \(\lambda= \sqrt{2}\) and \(\sqrt{3}\).