On rationally parametrized modular equations.

*(English)*Zbl 1214.11049If the modular curve \(X_0(M)\) associated with the modular group \(\Gamma_0(M)\) is of genus \(0\), then its function field is generated by a function \(t_M(\tau)\). This function \(t_M\) is called a Hauptmodul of \(\Gamma_0(M)\). Further, for an integer \(N\), if \(X_0(MN)\) is of genus \(0\), then \(t_M\) and \(t_M'=t_M(N\tau)\) are rationally expressed by a Hauptmodul \(t_{MN}\) of \(\Gamma_0(MN)\). By this fact, the author obtains the level \(N\) modular equation between \(t_M\) and \(t_M'\) rationally parametrized by \(t_{MN}\). Especially in the case \(M=1\) and \(t_1=j\), for every integer \(N\) such that \(X_0(N)\) is of genus \(0\), a rational parameterization of the classical modular equation of level \(N\) is explicitly computed. For example, the classical modular equation of level \(2\) has a parameterization: \(j=(t_2+16)^3/t_2,j'=j(2\tau)=(t_2+256)^3/t_2^2\). He defines functions \(h_M(t_M)\) of \(t_M\) which are solutions of Picard-Fuchs equation, of hypergeometric, Heun or more general type, which are considered to be periods of elliptic curve parametrized by \(t_M\). As a function of \(\tau\), \(\mathfrak h(\tau)=h_M(t_M(\tau))\) is a modular form of weight one with respect to \(\Gamma_0(M)\). From the parametrized equation of \(t_M\) and \(t_M'\) by \(t_{MN}\), the author obtains various modular equations (algebraic transformations) of \(h_M\) by using pullbacks of Picard-Fuchs equations along the map: \(X_0(MN)\rightarrow X_0(M)\) (viz. \(t_{MN}\mapsto t_M\)). The modular transformation of Ramanujan’s elliptic integrals \(K_r\) of signatures \(r=2,3,4,6\) are also obtained among these modular equations. The author points out that for \(r=2,3,4\), this gives a modern interpretation to Ramanujan’s theories of integrals to alternative bases and that his theory of signature \(6\) turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework. The functions \(t_M\), \(h_M\) and \(\mathfrak h_M(\tau)\) are given by eta products and the hypergeometric function \(_2F_1\) and their explicit forms and modular equations are listed in 19 tables.

Reviewer: Noburo Ishii (Osaka)