Ramanujan's modular equations and Atkin-Lehner involutions
This paper concerns a set of identities of which the following is the simplest: \[ -Q+{1\over Q}= P+{8\over P}, \] where \[ P= {\eta^3(\tau) \eta^3(3\tau)\over \eta^3(2\tau) \eta^3(6\tau)}\quad\text{and}\quad Q= {\eta^6(\tau) \eta^6(6\tau)\over \eta^6(2\tau) \eta^6(3\tau)}. \] Here \(\eta\) is the Dedekind \(\eta\)-function. The paper offers about a dozen identities of this shape, together with a unified method of proof, given in \S 4: ``Let \(\sigma\) be a modular function with invariance group \(A\) such that the genus \(g(A\setminus \mathbb{H}^*)\neq 0\). If there exists a group \(G\) such that \(m:= [G: A]<\infty\), \(g(G\setminus \mathbb{H}^*)= 0\), and \(A\vartriangleleft G\), then we always have an identity of the form \[ \sum^m_{i= 1}\sigma|_{g_i}= {n(f)\over d(f)}, \] where \(G= \bigcup^m_{i= 1} g_iA\), \(n(x)\) and \(d(x)\) are both polynomials in \(x\), and \(f\) generates the function field of \(G\setminus \mathbb{H}^*\). To determine \(d(f)\), we first set \(\wp:= \left\{\text{poles of }\sum^m_{i= 1} \sigma|_{g_i}\right\}\setminus \{\text{poles of }f\}\). Since by assumption, \(f\) is a bijection from \(G\setminus \mathbb{H}\to \mathbb{C}\cup\{\infty\}\), we conclude that \[ d(x)= \prod_{p\in\wp}(x- f(p))^{e_p}, \] where \[ e_p=- {\text{order of }(\sum^n_{i= 1}\sigma|_{g_i})\text{ at }p\over \text{order of }(f- f(p))\text{ at }p}. \] Note that \(d(x)\) is defined to be \(1\) if \(\wp= \phi\). The polynomial \(n(x)\) can then be determined by comparing the Fourier expansions of \(d(f)\sum^m_{i= 1}\sigma|_{g_i}\) and \(f\) at \(\infty\).'' In practice, \(G\) is an extension of a level \(N\) congruence subgroup by an Atkin-Lehner involution \(N_e= \left(\begin{smallmatrix} ae & b\\ cN & de\end{smallmatrix}\right)\), where \(e\| N\). This new method of proof yields some modular equations beyond those of Ramanujan. The final section lists ``all genus \(0\) discrete groups \(\Gamma\), \(\Gamma_0(N)\subset \Gamma\subset \Gamma_0(N)+\), where the \(\Gamma_0(N)+\) is generated by \(\Gamma_0(N)\) together with all its Atkin-Lehner involutions''.
scientific article; zbMATH DE number 1250636