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Zbl 0922.11040
Chan, Heng Huat; Lang, Mong Lung
Ramanujan's modular equations and Atkin-Lehner involutions.
(English)
[J] Isr. J. Math. 103, 1-16 (1998). ISSN 0021-2172; ISSN 1565-8511/e

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 $g(A\setminus \bbfH^*)\ne 0$. If there exists a group $G$ such that $m:= [G: A]<\infty$, $g(G\setminus \bbfH^*)= 0$, 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 $G\setminus \bbfH^*$. 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 $G\setminus \bbfH\to \bbfC\cup\{\infty\}$, 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 $\infty$.''\par In practice, $G$ is an extension of a level $N$ congruence subgroup by an Atkin-Lehner involution $N_e= \left(\smallmatrix ae & b\\ cN & de\endsmallmatrix\right)$, where $e\| N$. This new method of proof yields some modular equations beyond those of Ramanujan.\par 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''.
[M.Sheingorn (New York)]
MSC 2000:
*11F11 Modular forms, one variable
11F03 Modular and automorphic functions
11F20 Dedekind eta function, Dedekind sums

Keywords: Ramanujan modular equations; modular function; congruence subgroup; Atkin-Lehner involution

Cited in: Zbl 0973.11050

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