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Ramanujan’s modular equations and Atkin-Lehner involutions. (English) Zbl 0922.11040

This paper concerns a set of identities of which the following is the simplest:

$-Q+\frac{1}{Q}=P+\frac{8}{P},$

where

$P=\frac{{\eta }^{3}\left(\tau \right){\eta }^{3}\left(3\tau \right)}{{\eta }^{3}\left(2\tau \right){\eta }^{3}\left(6\tau \right)}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}Q=\frac{{\eta }^{6}\left(\tau \right){\eta }^{6}\left(6\tau \right)}{{\eta }^{6}\left(2\tau \right){\eta }^{6}\left(3\tau \right)}·$

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 §4: “Let $\sigma$ be a modular function with invariance group $A$ such that the genus $g\left(A\setminus {ℍ}^{*}\right)\ne 0$. If there exists a group $G$ such that $m:=\left[G:A\right]<\infty$, $g\left(G\setminus {ℍ}^{*}\right)=0$, and $A◃G$, then we always have an identity of the form

$\sum _{i=1}^{m}{\sigma |}_{{g}_{i}}=\frac{n\left(f\right)}{d\left(f\right)},$

where $G={\bigcup }_{i=1}^{m}{g}_{i}A$, $n\left(x\right)$ and $d\left(x\right)$ are both polynomials in $x$, and $f$ generates the function field of $G\setminus {ℍ}^{*}$. To determine $d\left(f\right)$, we first set $\wp :=\left\{\text{poles}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}{\sum }_{i=1}^{m}{\sigma |}_{{g}_{i}}\right\}\setminus \left\{\text{poles}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}f\right\}$. Since by assumption, $f$ is a bijection from $G\setminus ℍ\to ℂ\cup \left\{\infty \right\}$, we conclude that

$d\left(x\right)=\prod _{p\in \wp }{\left(x-f\left(p\right)\right)}^{{e}_{p}},$

where

${e}_{p}=-\frac{\text{order}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\left({\sum }_{i=1}^{n}\sigma {|}_{{g}_{i}}\right)\phantom{\rule{4.pt}{0ex}}\text{at}\phantom{\rule{4.pt}{0ex}}p}{\text{order}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\left(f-f\left(p\right)\right)\phantom{\rule{4.pt}{0ex}}\text{at}\phantom{\rule{4.pt}{0ex}}p}·$

Note that $d\left(x\right)$ is defined to be 1 if $\wp =\phi$. The polynomial $n\left(x\right)$ can then be determined by comparing the Fourier expansions of $d\left(f\right){\sum }_{i=1}^{m}{\sigma |}_{{g}_{i}}$ and $f$ at $\infty$.”

In practice, $G$ is an extension of a level $N$ congruence subgroup by an Atkin-Lehner involution ${N}_{e}=\left(\begin{array}{cc}ae& b\\ cN& de\end{array}\right)$, where $e\parallel N$. This new method of proof yields some modular equations beyond those of Ramanujan.

The final section lists “all genus 0 discrete groups ${\Gamma }$, ${{\Gamma }}_{0}\left(N\right)\subset {\Gamma }\subset {{\Gamma }}_{0}\left(N\right)+$, where the ${{\Gamma }}_{0}\left(N\right)+$ is generated by ${{\Gamma }}_{0}\left(N\right)$ together with all its Atkin-Lehner involutions”.

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11F03 Modular and automorphic functions 11F20 Dedekind eta function, Dedekind sums
##### References:
 [1] B. C. Berndt,Ramanujan Notebooks Part III, Springer-Verlag, New York, 1991. [2] B. C. Berndt,Ramanujan Notebooks Part IV, Springer-Verlag, New York, 1994. [3] B. C. Berndt and H. H. Chan,Some values for the Rogers-Ramanujan continued fraction, Canadian Journal of Mathematics47 (1995), 897–914. · doi:10.4153/CJM-1995-046-5 [4] B. C. Berndt, H. H. Chan and L.-C. Zhang,Ramanujan’s class invariants, Kronecker’s limit formula and modular equations, Transactions of the American Mathematical Society349 (1997), 2125–2173. · Zbl 0885.11058 · doi:10.1090/S0002-9947-97-01738-8 [5] B. C. Berndt, S. Bhargava and F. G. Garvan,Ramanujan’s theories of elliptic functions to alternative bases, Transactions of the American Mathematical Society347 (1995), 4163–4244. · Zbl 0843.33012 · doi:10.2307/2155035 [6] K. Harada,Modular functions, Modular forms and Finite Groups, Lecture Notes at The Ohio State University, 1987. [7] P. G. Kluit,On the normalizer of ${\Gamma }$ 0(N), inModular Function of One Variable V, Lecture Notes in Mathematics601, Springer-Verlag, Berlin, 1977. [8] M. I. Knopp,Modular Functions in Analytic Number Theory, Chelsea, New York, 1993. [9] T. Kondo,The automorphism group of Leech lattice and elliptic modular functions, Journal of the Mathematical Society of Japan37 (1985), 337–362. · Zbl 0572.10024 · doi:10.2969/jmsj/03720337 [10] M. Newman,Construction and application of a class of modular functions II, Proceedings of the London Mathematical Society9 (1959), 373–387. · Zbl 0178.43001 · doi:10.1112/plms/s3-9.3.373 [11] S. Ramanujan,Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.