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On rationally parametrized modular equations. (English) Zbl 1214.11049
If 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.

##### MSC:
 11F03 Modular and automorphic functions 11F20 Dedekind eta function, Dedekind sums 33C05 Classical hypergeometric functions, $${}_2F_1$$
##### Keywords:
modular equation; Ramanujan; elliptic integral
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